Your FREE AP Statistics Practice Test 2026 – 230+ Q&A

AP Statistics Practice Questions

1. Which measure of center is most resistant to the influence of extreme values in a data set?
   • The mean
   • The range
   • The median
   • The standard deviation

   Correct answer: The median

   The median is the most resistant measure of center because it depends only on the position of the middle value, not the magnitude of every observation. A single extreme value pulls the mean toward it but barely shifts the median. The range and standard deviation describe spread, not center, and both are strongly affected by extremes.

2. In a distribution that is skewed to the right, how does the mean typically compare to the median?
   • The mean is less than the median
   • The mean is greater than the median
   • The mean equals the median
   • The mean is always exactly twice the median

   Correct answer: The mean is greater than the median

   In a right-skewed distribution the mean is greater than the median because the long right tail contains large values that pull the mean upward while leaving the median near the bulk of the data. This relationship is reversed for left-skewed data, where the mean is pulled below the median. The mean and median coincide only in a symmetric distribution.

3. According to the empirical rule, approximately what percentage of values in a normal distribution fall within two standard deviations of the mean?
   • 95%
   • 68%
   • 99.7%
   • 50%

   Correct answer: 95%

   The empirical rule states that about 95% of values lie within two standard deviations of the mean in a normal distribution. About 68% fall within one standard deviation and about 99.7% within three. The value 50% corresponds to one side of the symmetric curve, not a standard-deviation interval.

4. A student scores 82 on a test with a mean of 74 and a standard deviation of 4. What is the student's z-score?
   • 0.5
   • 2.0
   • 8.0
   • -2.0

   Correct answer: 2.0

   The z-score of 2.0 is found by subtracting the mean from the value and dividing by the standard deviation: (82 - 74) / 4 = 8 / 4 = 2.0. This tells us the score is two standard deviations above the mean. A positive result confirms the score is above average, ruling out the negative option.

5. Using the 1.5 x IQR rule, an observation is flagged as an outlier if it falls below which boundary?
   • The mean minus two standard deviations
   • Q1 minus 1.5 times the IQR
   • Q1 minus the IQR
   • The median minus 1.5 times the IQR

   Correct answer: Q1 minus 1.5 times the IQR

   The lower outlier boundary is Q1 minus 1.5 times the IQR; any value below it is considered an outlier. The upper fence is Q3 plus 1.5 times the IQR. The rule is built on quartiles and the interquartile range, not on the mean, standard deviation, or median.

6. A data set has Q1 = 30 and Q3 = 50. What is the interquartile range?
   • 40
   • 80
   • 10
   • 20

   Correct answer: 20

   The interquartile range is 20, calculated as Q3 minus Q1, or 50 - 30 = 20. The IQR measures the spread of the middle 50% of the data. Adding the quartiles (80) or averaging them (40) does not produce a measure of spread.

7. Which five values make up the five-number summary of a quantitative data set?
   • Mean, median, mode, range, standard deviation
   • Minimum, Q1, median, Q3, maximum
   • Minimum, mean, median, mode, maximum
   • Q1, Q2, Q3, mean, standard deviation

   Correct answer: Minimum, Q1, median, Q3, maximum

   The five-number summary consists of the minimum, first quartile, median, third quartile, and maximum. These five values describe center, spread, and the extremes of a distribution and are the basis for constructing a boxplot. The mean, mode, and standard deviation are not part of the five-number summary.

8. What does the length of the box in a boxplot directly represent?
   • The total range of the data
   • The standard deviation
   • The interquartile range
   • The mean of the data

   Correct answer: The interquartile range

   The length of the box in a boxplot represents the interquartile range, the distance from Q1 to Q3 spanning the middle 50% of the data. The whiskers, not the box, extend toward the extremes. The boxplot does not display the mean or standard deviation.

9. A symmetric, bell-shaped distribution of a continuous quantitative variable is best described as which type?
   • Normal distribution
   • Uniform distribution
   • Bimodal distribution
   • Skewed distribution

   Correct answer: Normal distribution

   A symmetric, bell-shaped distribution is described as a normal distribution, the foundation for z-scores and the empirical rule. A uniform distribution is flat, a bimodal distribution has two peaks, and a skewed distribution is asymmetric with a longer tail on one side.

10. The heights of adult women are approximately normal with a mean of 64 inches and a standard deviation of 2.5 inches. Between which two heights do about 68% of these women fall?
    • 61.5 inches and 66.5 inches
    • 59 inches and 69 inches
    • 56.5 inches and 71.5 inches
    • 62.5 inches and 65.5 inches

    Correct answer: 61.5 inches and 66.5 inches

    About 68% of the women fall between 61.5 and 66.5 inches, found by taking the mean plus or minus one standard deviation: 64 - 2.5 = 61.5 and 64 + 2.5 = 66.5. The empirical rule assigns 68% to one standard deviation. The interval 59 to 69 covers two standard deviations (about 95%).

11. If every value in a data set is increased by 10, what happens to the standard deviation?
    • It increases by 10
    • It increases by $\sqrt{10}$
    • It stays the same
    • It doubles

    Correct answer: It stays the same

    The standard deviation stays the same because adding a constant shifts every value by the same amount, leaving the distances between values unchanged. Standard deviation measures spread, which is unaffected by a uniform shift. Only the center, such as the mean, increases by 10.

12. A normally distributed exam has a mean of 500. Student A has a z-score of 1.5 and Student B has a z-score of -0.5. Which statement correctly compares their raw scores?
    • Student B scored higher than Student A
    • Student A scored higher than Student B
    • Both students scored exactly 500
    • The z-scores do not allow any comparison of their scores

    Correct answer: Student A scored higher than Student B

    Student A scored higher than Student B because a z-score measures how many standard deviations a value lies above or below the mean. Student A is 1.5 standard deviations above the mean while Student B is half a standard deviation below it. Since both come from the same distribution, the larger z-score corresponds to the higher raw score.

13. A real estate report states that the mean home price in a town is far higher than the median home price. What does this most strongly suggest about the distribution of home prices?
    • It is skewed to the left
    • It is perfectly symmetric
    • It is uniform
    • It is skewed to the right

    Correct answer: It is skewed to the right

    The distribution is most likely skewed to the right because the mean being much larger than the median indicates a long tail of expensive homes pulling the mean upward. In a left-skewed distribution the mean would fall below the median, and in a symmetric distribution the two measures would be roughly equal.

14. A data set of daily temperatures has Q1 = 58, Q3 = 70, and includes a reading of 92 degrees. Using the 1.5 x IQR rule, is 92 an outlier?
    • No, because it is below 88
    • Yes, because it exceeds 70
    • Yes, because it exceeds 88
    • No, because it is within two standard deviations

    Correct answer: Yes, because it exceeds 88

    The value 92 is an outlier because it exceeds the upper fence of 88. The IQR is 70 - 58 = 12, so the upper boundary is Q3 + 1.5 times IQR = 70 + 18 = 88. Since 92 is greater than 88, it is flagged as a high outlier. The 1.5 x IQR rule uses quartiles, not standard deviations.

15. Two boxplots compare test scores from two classes. Class X has a longer box than Class Y, but both have the same median. What can be concluded?
    • Class Y has greater variability in the middle 50% of scores
    • The two classes have identical distributions
    • Class X has greater variability in the middle 50% of scores
    • Class X has a higher typical score than Class Y

    Correct answer: Class X has greater variability in the middle 50% of scores

    Class X has greater variability in the middle 50% of scores because a longer box means a larger interquartile range, indicating more spread among the central half of the data. The equal medians mean the typical scores are comparable, so neither class is centered higher. A longer box reflects spread, not a higher center.

16. Which of the following describes the shape of a distribution whose histogram has a long tail extending toward the lower values on the left?
    • Skewed to the right
    • Symmetric
    • Uniform
    • Skewed to the left

    Correct answer: Skewed to the left

    A distribution with a long tail extending toward the lower values is skewed to the left, sometimes called negatively skewed. The direction of skew is named for the side where the tail stretches, not where the peak sits. A right-skewed distribution would have its tail toward the higher values, and a symmetric distribution would have no tail.

17. A teacher reports that a class data set has a mean of 75 and a standard deviation of 0. What must be true about the data?
    • Every value in the data set is 75
    • The values are evenly spread from 0 to 75
    • Half the values are above 75 and half below
    • The data set contains exactly one outlier

    Correct answer: Every value in the data set is 75

    Every value in the data set must equal 75 because a standard deviation of 0 means there is no spread at all and all observations are identical. Standard deviation measures how far values deviate from the mean, so zero deviation forces every value to match the mean exactly. Any variation would produce a positive standard deviation.

18. A scholarship requires applicants to score at least 1.0 standard deviation above the mean on a normally distributed admissions test. Using the empirical rule, approximately what percentage of test-takers qualify?
    • About 16%
    • About 32%
    • About 68%
    • About 5%

    Correct answer: About 16%

    About 16% of test-takers qualify because the empirical rule places about 68% of values within one standard deviation of the mean, leaving about 32% in the two tails combined. Since the distribution is symmetric, half of that 32%, or about 16%, lies in the upper tail beyond one standard deviation above the mean.

19. A data set's five-number summary is 12, 20, 25, 28, 60. Which feature of this summary most strongly signals a possible high outlier?
    • The median of 25 is close to Q3 of 28
    • The minimum of 12 is below the median
    • The first quartile of 20 is below the median
    • The maximum of 60 is far above Q3 of 28

    Correct answer: The maximum of 60 is far above Q3 of 28

    The maximum of 60 sitting far above Q3 of 28 most strongly signals a possible high outlier. The IQR is 28 - 20 = 8, so the upper fence is 28 + 1.5 times 8 = 40, and 60 exceeds it. The median being near Q3 and the quartiles falling on either side of the median are normal features, not outlier signals.

20. Which type of graph is used to display the relationship between two quantitative variables?
    • A scatterplot
    • A boxplot
    • A bar chart
    • A histogram

    Correct answer: A scatterplot

    A scatterplot is the correct display because it plots paired values of two quantitative variables as points, letting you assess the form, direction, and strength of their association. A boxplot and a histogram each display only a single quantitative variable, and a bar chart displays counts for a categorical variable.

21. The correlation coefficient r can take on values only within which range?
    • From 0 to 1
    • From 0 to 100
    • From -1 to 1
    • From -100 to 100

    Correct answer: From -1 to 1

    The correlation coefficient is bounded between -1 and 1 inclusive. A value near -1 or 1 indicates a strong linear relationship, while a value near 0 indicates a weak linear relationship; the sign shows direction. The ranges 0 to 1, 0 to 100, and -100 to 100 misstate these limits.

22. A least-squares regression line predicting weight from height has the equation y-hat = 50 + 4x, where x is height in inches and y is weight in pounds. How is the slope interpreted in context?
    • For each additional pound of weight, height increases by 4 inches
    • For each additional inch of height, predicted weight increases by 4 pounds
    • A person with zero height is predicted to weigh 4 pounds
    • For each additional inch of height, predicted weight increases by 50 pounds

    Correct answer: For each additional inch of height, predicted weight increases by 4 pounds

    The slope of 4 means that for each additional inch of height, the predicted weight increases by 4 pounds. The slope describes the predicted change in the response variable (weight) per one-unit increase in the explanatory variable (height), not the reverse. The value 50 is the y-intercept, the predicted weight when height is zero, not the slope.

23. A least-squares regression model reports a coefficient of determination of $r^2 = 0.81$. What does this value mean?
    • The slope of the regression line is 0.81
    • About 81% of the data points fall exactly on the regression line
    • The correlation between the variables is exactly 0.81
    • About 81% of the variation in the response variable is explained by the linear model

    Correct answer: About 81% of the variation in the response variable is explained by the linear model

    An $r^2$ of 0.81 means about 81% of the variation in the response variable is explained by the least-squares model relating it to the explanatory variable. It is the square of the correlation, so r would be about 0.9, not 0.81. It does not represent the slope and does not mean 81% of points lie on the line.

24. When a residual plot for a linear regression shows a clear curved (U-shaped) pattern rather than random scatter, what does this most strongly suggest?
    • The two variables have no association at all
    • A linear model is not appropriate for the data
    • The correlation coefficient must equal zero
    • The data contain no influential points

    Correct answer: A linear model is not appropriate for the data

    A curved pattern in the residual plot most strongly suggests that a linear model is not appropriate, because well-fitting linear models produce residuals scattered randomly around zero with no leftover pattern. A clear curve means systematic structure remains unexplained by the line. The pattern does not imply zero correlation or rule out influential points.

25. In a simple random sample of size n, what is true about every possible group of n individuals from the population?
    • Each possible group of n individuals has an equal chance of being selected
    • Only groups that represent every subgroup can be selected
    • Larger groups are more likely to be selected than smaller ones
    • Each individual is selected based on convenience to the researcher

    Correct answer: Each possible group of n individuals has an equal chance of being selected

    The defining property of a simple random sample is that every possible group of n individuals from the population has an equal chance of being chosen. This is stronger than merely giving each individual an equal chance. Selecting by convenience or forcing subgroup representation describes other, non-SRS methods.

26. A pollster divides voters into age groups and then takes a separate simple random sample from within each age group. Which sampling method is being used?
    • Convenience sampling
    • Cluster sampling
    • Stratified random sampling
    • Systematic sampling

    Correct answer: Stratified random sampling

    This is stratified random sampling because the population is split into homogeneous groups, called strata, and a random sample is drawn from within each one. The strata here are age groups, chosen because members within them are similar. Cluster sampling would instead sample whole groups, and convenience sampling would pick whoever is easiest to reach.

27. A researcher randomly selects several entire classrooms in a school and surveys every student in those chosen classrooms. Which sampling method does this describe?
    • Cluster sampling
    • Stratified random sampling
    • Simple random sampling
    • Convenience sampling

    Correct answer: Cluster sampling

    This describes cluster sampling because the population is divided into groups, called clusters, and entire randomly selected clusters are surveyed in full. Each classroom is a cluster meant to resemble the population as a whole. This differs from stratified sampling, where only some members are sampled from every group rather than all members from a few groups.

28. A student surveys friends in the cafeteria because they are easy to reach, then generalizes the results to the whole school. What is the primary weakness of this approach?
    • It requires too large a sample size
    • It guarantees an overestimate of the true value
    • It is a stratified sample that ignores clusters
    • It is a convenience sample that is likely biased

    Correct answer: It is a convenience sample that is likely biased

    The primary weakness is that this is a convenience sample, which is likely biased because the friends chosen may differ systematically from the rest of the school. Selecting whoever is easiest to reach gives no assurance the sample represents the population. The flaw is bias from non-random selection, not sample size, and the direction of any error cannot be guaranteed.

29. In a well-designed experiment, what is the main purpose of including a control group?
    • To increase the total number of subjects in the study
    • To provide a baseline for comparison against the treatment group
    • To guarantee that no confounding variables exist
    • To ensure the sample is selected at random

    Correct answer: To provide a baseline for comparison against the treatment group

    The control group's main purpose is to provide a baseline for comparison, so the effect of the treatment can be judged against subjects who did not receive it. Without this comparison, researchers cannot tell whether changes are due to the treatment. A control group does not enlarge the sample's reach or by itself eliminate every confounding variable.

30. Which study design allows researchers to draw a cause-and-effect conclusion between an explanatory variable and a response?
    • An observational study with a large sample
    • A survey using stratified random sampling
    • A randomized comparative experiment
    • A study based on voluntary response data

    Correct answer: A randomized comparative experiment

    A randomized comparative experiment is the design that supports a cause-and-effect conclusion, because randomly assigning treatments balances out other variables across groups. Observational studies and surveys, even with excellent sampling, can establish association but not causation because lurking variables are not controlled. Voluntary response data is especially prone to bias.

31. In an experiment testing a new fertilizer, the amount of sunlight different plots receive also varies and is tied to which fertilizer they got. Sunlight in this study is best described as what?
    • A control group
    • A response variable
    • A stratifying variable
    • A confounding variable

    Correct answer: A confounding variable

    Sunlight is a confounding variable because its effect on plant growth cannot be separated from the effect of the fertilizer, since the two vary together. This makes it impossible to attribute results to the fertilizer alone. It is not the response being measured, nor a control group, nor a basis for forming strata.

32. What is the most important benefit of using random assignment of treatments in an experiment?
    • It guarantees the sample looks exactly like the population
    • It tends to create groups that are similar except for the treatment
    • It removes the need to have a control group
    • It allows the results to be generalized to a larger population

    Correct answer: It tends to create groups that are similar except for the treatment

    Random assignment tends to create treatment groups that are roughly similar in all other variables, so any difference in outcomes can be attributed to the treatment itself. This is what permits causal conclusions. It does not make the sample mirror the population, that is the job of random sampling, which governs generalization instead.

33. A nutrition researcher records what 500 randomly selected adults already eat and tracks their health over time, without assigning any diets. The researcher finds that coffee drinkers have fewer headaches. What is the strongest valid conclusion?
    • There is an association between coffee drinking and fewer headaches
    • Drinking coffee causes fewer headaches
    • Random assignment removed all confounding variables
    • The study proves coffee should be recommended for headaches

    Correct answer: There is an association between coffee drinking and fewer headaches

    The strongest valid conclusion is that there is an association between coffee drinking and fewer headaches, because this is an observational study with no treatment assigned. Without random assignment, lurking variables such as overall lifestyle could explain the link, so causation cannot be claimed. Only a randomized experiment could support a cause-and-effect statement.

34. A factory wants to inspect bolts coming off an assembly line and selects every 25th bolt for testing. To best apply a simple random sample instead, the inspector should do which of the following?
    • Continue selecting every 25th bolt but start earlier in the day
    • Inspect only the bolts that are easiest to reach on the line
    • Inspect all bolts from the first production batch
    • Number all bolts produced and use a random process to choose which to inspect

    Correct answer: Number all bolts produced and use a random process to choose which to inspect

    To take a simple random sample, the inspector should number all bolts and use a random process so that every possible group of bolts is equally likely to be chosen. Selecting every 25th item is systematic sampling, not an SRS. Choosing the easiest-to-reach bolts is convenience sampling, and inspecting one batch in full is a single cluster.

35. A school administrator wants opinions from students and ensures that freshmen, sophomores, juniors, and seniors are each represented proportionally by randomly sampling within each grade. Why might this stratified approach be preferred over a single simple random sample?
    • It removes the need for any randomization
    • It guarantees a cause-and-effect conclusion
    • It ensures each grade level is adequately represented and can reduce variability
    • It is the same as taking a convenience sample of each grade

    Correct answer: It ensures each grade level is adequately represented and can reduce variability

    Stratifying by grade ensures each grade level is adequately represented and can reduce the variability of estimates when grades differ in opinion. A single random sample might by chance underrepresent a grade. The method still uses randomization within strata and, being a survey, cannot establish causation.

36. A medical team tests a drug by giving half the volunteers the real drug and half an identical-looking pill with no active ingredient, with no one knowing which they received. The purpose of the identical-looking inactive pill is best described as what?
    • A confounding variable introduced on purpose
    • A placebo serving as a control treatment
    • A stratifying factor for the volunteers
    • A method of cluster sampling the volunteers

    Correct answer: A placebo serving as a control treatment

    The inactive pill is a placebo serving as a control treatment, allowing a fair comparison by accounting for the psychological effect of simply taking a pill. The control group receiving it provides the baseline against which the real drug is judged. It is not a confounding variable, a basis for strata, or a sampling technique.

37. Two researchers study whether exercise lowers blood pressure. Researcher A randomly assigns volunteers to an exercise program or a no-exercise group. Researcher B simply compares people who already exercise to those who do not. Which statement correctly contrasts what each can conclude?
    • Both can conclude exercise causes lower blood pressure
    • Researcher B can claim causation while Researcher A can only show association
    • Neither can show any relationship between exercise and blood pressure
    • Researcher A can claim causation while Researcher B can only show association

    Correct answer: Researcher A can claim causation while Researcher B can only show association

    Researcher A can claim causation while Researcher B can only show association, because random assignment in Researcher A's experiment balances out lurking variables, isolating exercise as the cause. Researcher B's observational comparison leaves confounders, such as diet, uncontrolled. The presence or absence of random assignment is what separates a causal claim from a merely associational one.

38. A city surveys residents about a new park by mailing forms and using only the responses from people who chose to mail them back. Why is the resulting data likely to be misleading?
    • Voluntary response tends to overrepresent those with strong opinions
    • The sample is too large to analyze accurately
    • Mailing forms is a form of cluster sampling
    • Random assignment was not used to mail the forms

    Correct answer: Voluntary response tends to overrepresent those with strong opinions

    The data is likely misleading because voluntary response tends to overrepresent people with strong opinions, who are more motivated to reply. This produces a biased picture of the community's views rather than a representative one. The problem is self-selection bias, not sample size, and random assignment applies to experiments, not to fixing this survey flaw.

39. A random variable X represents the number of successes in a fixed number of independent trials, each with the same probability of success. Which type of probability distribution does X follow?
    • A geometric distribution
    • A uniform distribution
    • A binomial distribution
    • A normal distribution

    Correct answer: A binomial distribution

    X follows a binomial distribution because it counts the number of successes in a fixed number of independent trials, each having two outcomes and the same success probability. A geometric distribution instead counts trials until the first success, with no fixed number of trials. A uniform distribution assigns equal probability to outcomes, and a normal distribution is continuous rather than a count.

40. Two events A and B are described as mutually exclusive. What does this mean?
    • Knowing A occurred does not change the probability of B
    • Events A and B cannot both occur at the same time
    • Events A and B must both occur together
    • The probability of A equals the probability of B

    Correct answer: Events A and B cannot both occur at the same time

    Mutually exclusive means events A and B cannot both occur at the same time, so they share no common outcomes and their joint probability is zero. This is a statement about overlap, not about whether one event changes the other's probability, which describes independence. Mutual exclusivity does not require the two events to have equal probabilities.

41. What does the expected value of a discrete random variable represent?
    • The most likely single outcome of the variable
    • The largest value the variable can take
    • The long-run average value of the variable over many repetitions
    • The probability that the variable equals zero

    Correct answer: The long-run average value of the variable over many repetitions

    The expected value represents the long-run average value of the random variable if the random process were repeated many times. It is computed as the sum of each value times its probability and need not equal any single attainable outcome. It is not the most likely value, the maximum value, or a probability.

42. A geometric random variable counts which of the following?
    • The number of trials needed to get the first success
    • The number of successes in a fixed number of trials
    • The proportion of successes in a large sample
    • The total number of possible outcomes in an experiment

    Correct answer: The number of trials needed to get the first success

    A geometric random variable counts the number of independent trials needed to obtain the first success, where each trial has the same success probability. This contrasts with a binomial random variable, which counts successes within a fixed number of trials. It is a count of trials, not a proportion or a tally of all possible outcomes.

43. A fair six-sided die is rolled. Let event A be rolling an even number and event B be rolling a 5. Are events A and B mutually exclusive?
    • No, because 5 is an even number
    • Yes, because no even number is also a 5
    • No, because both events involve the same die
    • Yes, because the events have equal probability

    Correct answer: Yes, because no even number is also a 5

    Events A and B are mutually exclusive because no outcome belongs to both: 5 is odd, so it can never be one of the even numbers 2, 4, or 6. With no shared outcome, the two events cannot occur on the same roll. Sharing the same die or having any particular probabilities does not determine mutual exclusivity.

44. A multiple-choice quiz has 10 questions, each with 4 choices, and a student guesses every answer at random. What is the expected number of correct answers?
    • 4
    • 10
    • 2.5
    • 5

    Correct answer: 2.5

    The expected number correct is 2.5, found for a binomial setting as the number of trials times the success probability: 10 times $\frac{1}{4}$ equals 2.5. Each question has a $\frac{1}{4}$ chance of a correct guess, and expectation adds these across all 10 questions. The values 4, 5, and 10 ignore the correct multiplication of trials by probability.

45. Events A and B are independent, with P(A) = 0.5 and P(B) = 0.4. What is the probability that both A and B occur?
    • 0.90
    • 0.10
    • 0.45
    • 0.20

    Correct answer: 0.20

    The probability that both occur is 0.20 because for independent events the joint probability equals the product of the individual probabilities: 0.5 times 0.4 equals 0.20. The multiplication rule applies only because the events are independent. Adding the probabilities (0.90) or other combinations does not give the probability of both events happening.

46. In a class, 60% of students take Spanish and 30% of students take both Spanish and art. Given that a student takes Spanish, what is the probability that the student also takes art?
    • 0.30
    • 0.18
    • 0.90
    • 0.50

    Correct answer: 0.50

    The conditional probability is 0.50, found by dividing the probability of both events by the probability of the given event: 0.30 divided by 0.60 equals 0.50. Conditional probability restricts attention to the Spanish-takers and asks what fraction of them also take art. Simply reporting 0.30 ignores that the condition narrows the relevant group.

47. A basketball player makes 70% of her free throws, and each attempt is independent. What is the probability that she makes her first three free throws and misses the fourth?
    • 0.7 times 0.7 times 0.7 times 0.3
    • 0.7 times 0.3
    • 0.7 times 4
    • 0.3 times 0.3 times 0.3 times 0.7

    Correct answer: 0.7 times 0.7 times 0.7 times 0.3

    The probability is 0.7 times 0.7 times 0.7 times 0.3 because independent attempts let us multiply the probabilities of the specified outcomes: three makes at 0.7 each and one miss at 0.3. The multiplication rule for independent events applies to this exact sequence. Reversing the make and miss probabilities or adding terms does not match the described outcome.

48. A medical test is given to a population. Among people who actually have a disease, the test comes back positive 95% of the time. This 95% figure is best described as which kind of probability?
    • A conditional probability of testing positive given the disease is present
    • The probability that a randomly chosen person tests positive
    • A mutually exclusive probability
    • The expected value of a positive test

    Correct answer: A conditional probability of testing positive given the disease is present

    The 95% is a conditional probability of testing positive given that the disease is present, because it describes the chance of a positive result within the subgroup that actually has the disease. The condition restricts the population to diseased individuals before measuring the positive rate. It is not the overall positive rate, an expected value, or a statement about mutual exclusivity.

49. A salesperson has a 20% chance of closing each independent sales call. Let X be the number of calls up to and including the first successful close. What is the probability that the first success happens on exactly the third call?
    • 0.2 times 0.2 times 0.2
    • 3 times 0.2
    • 0.8 times 0.8 times 0.8
    • 0.8 times 0.8 times 0.2

    Correct answer: 0.8 times 0.8 times 0.2

    The probability is 0.8 times 0.8 times 0.2 because a geometric random variable requires the first two calls to be failures, each with probability 0.8, followed by a success with probability 0.2 on the third call. The independent trials let us multiply these in sequence. Using only success probabilities or multiplying by 3 misrepresents the required pattern of two failures then a success.

50. Suppose events A and B have P(A) = 0.4, P(B) = 0.5, and P(A and B) = 0.2. A student claims A and B are both mutually exclusive and independent. Which statement is correct?
    • They are mutually exclusive but not independent
    • They are independent but not mutually exclusive
    • They are both mutually exclusive and independent
    • They are neither mutually exclusive nor independent

    Correct answer: They are independent but not mutually exclusive

    The events are independent but not mutually exclusive. They are not mutually exclusive because P(A and B) equals 0.2, which is not zero, so the events can occur together. They are independent because the product P(A) times P(B), which is 0.4 times 0.5 equals 0.2, matches the actual joint probability. Mutually exclusive events with nonzero probabilities can never be independent.

51. A carnival game costs nothing to consider and pays $10 with probability 0.1, pays $2 with probability 0.3, and pays $0 with probability 0.6. What is the expected payout for one play?
    • $4.00
    • $1.60
    • $12.00
    • $2.00

    Correct answer: $1.60

    The expected payout is $1.60, computed by summing each payout times its probability: 10 times 0.1 plus 2 times 0.3 plus 0 times 0.6 equals 1 plus 0.6 plus 0, which is 1.60. Expected value weights every possible payout by how likely it is. Averaging the dollar amounts directly or adding them ignores the differing probabilities.

52. A quality inspector examines 8 independently produced parts, each defective with probability 0.05. Treating the count of defective parts as a random variable, why is a binomial model appropriate here rather than a geometric model?
    • Because there is a fixed number of trials and we are counting defectives among them
    • Because we are waiting for the first defective part to appear
    • Because the probability of a defect changes from part to part
    • Because the parts are not independent of one another

    Correct answer: Because there is a fixed number of trials and we are counting defectives among them

    A binomial model fits because there is a fixed number of trials, the 8 parts, and the random variable counts how many are defective among them. A geometric model would instead apply if we were counting parts until the first defective appeared, with no fixed total. The binomial conditions of a set number of independent trials with a constant success probability are all met here.

53. What does a sampling distribution describe?
    • The distribution of all individual values in a single sample
    • The distribution of values in the entire population
    • The distribution of a statistic computed from all possible samples of a given size
    • The distribution of the sample size across repeated studies

    Correct answer: The distribution of a statistic computed from all possible samples of a given size

    A sampling distribution is the distribution of a statistic computed from all possible samples of a given size. Each sample produces one value of the statistic (such as a sample mean or sample proportion), and collecting those values over all samples forms the sampling distribution. It is not the distribution of raw data within one sample, nor the distribution of the population itself, and it is built for a fixed sample size rather than describing how the sample size varies.

54. According to the central limit theorem, what happens to the shape of the sampling distribution of the sample mean as the sample size increases?
    • It becomes approximately normal regardless of the population's shape
    • It takes on the exact shape of the population distribution
    • It becomes increasingly skewed
    • It becomes uniform across all possible values

    Correct answer: It becomes approximately normal regardless of the population's shape

    The central limit theorem states the sampling distribution of the sample mean becomes approximately normal as the sample size increases, no matter the shape of the population distribution. This is why large samples allow normal-based inference even when the underlying population is skewed. It does not mirror the population shape, become more skewed, or turn uniform.

55. A population has mean 70 and standard deviation 12. For samples of size 36, what is the standard deviation of the sampling distribution of the sample mean?
    • 12
    • 6
    • 0.33
    • 2

    Correct answer: 2

    The standard deviation of the sampling distribution of the sample mean is 2, found by dividing the population standard deviation by $\sqrt{n}$: 12 divided by $\sqrt{36}$, which is 12 divided by 6. The value 12 ignores the sample-size adjustment entirely, while the other values come from dividing incorrectly.

56. What is the mean of the sampling distribution of the sample proportion equal to?
    • The sample size times the population proportion
    • The true population proportion
    • The population proportion divided by the sample size
    • Always 0.5

    Correct answer: The true population proportion

    The mean of the sampling distribution of the sample proportion equals the true population proportion, which is what makes the sample proportion an unbiased estimator. The center of the sampling distribution sits at the population value it estimates. It is not scaled by the sample size and is not fixed at 0.5.

57. A sample proportion is described as an unbiased estimator of the population proportion. What does this mean about its sampling distribution?
    • Every sample produces the exact population proportion
    • The sampling distribution has no variability
    • The sampling distribution is always perfectly normal
    • The mean of the sampling distribution equals the population proportion

    Correct answer: The mean of the sampling distribution equals the population proportion

    Saying the sample proportion is unbiased means the mean of its sampling distribution equals the population proportion, so on average it neither overestimates nor underestimates the target value. Individual samples still vary, so not every sample hits the population value, the distribution still has spread, and normality depends on conditions rather than being automatic.

58. A population is strongly right-skewed. A researcher wants the sampling distribution of the sample mean to be approximately normal. Based on the central limit theorem, what should the researcher do?
    • Take a sufficiently large sample so the distribution of the mean is approximately normal
    • Take a very small sample to better match the skewed population
    • Transform every individual value into a z-score before sampling
    • Conclude that normal-based inference is impossible for skewed populations

    Correct answer: Take a sufficiently large sample so the distribution of the mean is approximately normal

    The researcher should take a sufficiently large sample so the distribution of the mean is approximately normal, which is exactly what the central limit theorem guarantees even for a skewed population. A small sample would leave the sampling distribution skewed, standardizing individual values does not change the population's shape, and normal-based inference is achievable precisely because larger samples produce an approximately normal sampling distribution.

59. A factory's bolt lengths have mean 50 mm and standard deviation 4 mm. As the sample size used to compute sample means increases from 16 to 64, what happens to the standard deviation of the sampling distribution of the sample mean?
    • It stays the same because the population standard deviation does not change
    • It is cut in half
    • It is reduced to one-fourth
    • It doubles

    Correct answer: It is cut in half

    The standard deviation of the sampling distribution is cut in half. It equals the population standard deviation divided by $\sqrt{n}$; going from 16 to 64 multiplies the sample size by 4, and $\sqrt{4} = 2$, so the standard deviation is divided by 2. It does change with sample size, and it does not shrink to one-fourth or grow.

60. In a large population, 40% of voters support a measure. For random samples of 200 voters, the sampling distribution of the sample proportion is centered at what value?
    • 0.40
    • 0.50
    • 0.20
    • 80

    Correct answer: 0.40

    The sampling distribution of the sample proportion is centered at 0.40, the true population proportion, because the sample proportion is an unbiased estimator. The value 0.50 is a common default that ignores the given proportion, 0.20 misuses the figures, and 80 is the expected count of supporters rather than a proportion.

61. Two sampling distributions of the sample mean are built from the same population, one using samples of size 25 and the other using samples of size 100. How do these two sampling distributions compare?
    • The size-100 distribution has more variability than the size-25 distribution
    • The size-100 distribution has less variability and is more tightly clustered around the population mean
    • Both distributions have identical variability because they share a population
    • The size-25 distribution is centered at a higher value than the size-100 distribution

    Correct answer: The size-100 distribution has less variability and is more tightly clustered around the population mean

    The size-100 distribution has less variability and is more tightly clustered around the population mean, because the standard deviation of the sampling distribution decreases as sample size increases. Both distributions are centered at the same population mean, so neither is higher than the other, and their variability is not identical since larger samples produce smaller spread.

62. A student claims that because a population of reaction times is heavily skewed, the sampling distribution of the sample mean for samples of size 80 must also be heavily skewed. Why is this reasoning flawed?
    • The sampling distribution always exactly copies the population's shape
    • Skewed populations cannot be sampled at all
    • The central limit theorem makes the sampling distribution of the mean approximately normal for a large sample like 80
    • The sampling distribution of the mean is always uniform regardless of sample size

    Correct answer: The central limit theorem makes the sampling distribution of the mean approximately normal for a large sample like 80

    The reasoning is flawed because the central limit theorem makes the sampling distribution of the mean approximately normal for a large sample like 80, even when the population is skewed. The sampling distribution does not copy the population's shape at large sample sizes, skewed populations can certainly be sampled, and the sampling distribution is not uniform.

63. When setting up a significance test, what is the role of the null hypothesis?
    • It states a claim of no effect or no difference that the test attempts to find evidence against
    • It states the effect the researcher hopes to prove is true
    • It specifies the sample proportion observed in the data
    • It guarantees that any difference found is due to chance

    Correct answer: It states a claim of no effect or no difference that the test attempts to find evidence against

    The null hypothesis states a claim of no effect or no difference, and the test gathers evidence against it. It typically expresses that a population proportion equals a specific value, such as p = 0.5. The alternative hypothesis, not the null, states what the researcher suspects is true, and the null is about the population parameter rather than the observed sample proportion.

64. In a one-proportion significance test, what does the p-value measure?
    • The probability that the null hypothesis is true
    • The probability that the alternative hypothesis is false
    • The proportion of the population that supports the claim
    • The probability of observing a sample result at least as extreme as the one obtained, assuming the null hypothesis is true

    Correct answer: The probability of observing a sample result at least as extreme as the one obtained, assuming the null hypothesis is true

    The p-value measures the probability of getting a sample result at least as extreme as the observed one, calculated under the assumption that the null hypothesis is true. A small p-value means such a result would be unlikely if the null were correct, providing evidence against it. The p-value is not the probability that the null is true, nor a population proportion.

65. A confidence interval for a population proportion is constructed as the sample proportion plus or minus a margin of error. What two components are multiplied together to form that margin of error?
    • The sample size and the population proportion
    • The point estimate and the confidence level
    • A critical value and the standard error of the sample proportion
    • The p-value and the significance level

    Correct answer: A critical value and the standard error of the sample proportion

    The margin of error is the product of a critical value (a z* from the standard normal distribution for the chosen confidence level) and the standard error of the sample proportion. Together they set how far the interval extends on each side of the estimate. The margin of error does not multiply the sample size by the proportion, nor does it involve the p-value.

66. In the context of a significance test, what is a Type I error?
    • Rejecting the null hypothesis when it is actually true
    • Failing to reject the null hypothesis when it is actually false
    • Choosing too small a sample size for the test
    • Calculating the standard error incorrectly

    Correct answer: Rejecting the null hypothesis when it is actually true

    A Type I error is rejecting the null hypothesis when it is in fact true, meaning the test concludes there is an effect when there is none. Its probability equals the significance level alpha. Failing to reject a false null is instead a Type II error, and a Type I error is about an incorrect conclusion rather than a sample-size or calculation mistake.

67. What does the power of a significance test represent?
    • The probability of making a Type I error
    • The probability that the null hypothesis is true
    • The width of the confidence interval
    • The probability of correctly rejecting a false null hypothesis

    Correct answer: The probability of correctly rejecting a false null hypothesis

    The power of a test is the probability of correctly rejecting the null hypothesis when it is actually false, that is, the chance of detecting a real effect. Higher power means the test is better at finding true differences. It is not the probability of a Type I error, which is alpha, and it is unrelated to the width of a confidence interval.

68. A researcher tests whether more than half of voters support a measure. Which set of hypotheses correctly states this one-sided test?
    • H0: p = 0.5 versus Ha: p > 0.5
    • H0: p > 0.5 versus Ha: p = 0.5
    • H0: p-hat = 0.5 versus Ha: p-hat > 0.5
    • H0: p = 0.5 versus Ha: p < 0.5

    Correct answer: H0: p = 0.5 versus Ha: p > 0.5

    The correct hypotheses are H0: p = 0.5 versus Ha: p > 0.5, because the claim of more than half support is a one-sided alternative in the greater-than direction. Hypotheses are written about the population proportion p, not the sample proportion p-hat. The null always carries the equality, and the less-than alternative would test the opposite direction.

69. In a sample of 400 adults, 240 said they exercise weekly. What is the sample proportion used as the point estimate when building a confidence interval for the population proportion?
    • 0.40
    • 0.60
    • 240
    • 0.024

    Correct answer: 0.60

    The point estimate is the sample proportion 0.60, found by dividing the number of successes by the sample size: 240 divided by 400 equals 0.60. This value serves as the center of the confidence interval for the true population proportion. The count 240 is not a proportion, and 0.40 reverses the success and failure groups.

70. A 95% confidence interval for the proportion of residents who recycle is found to be 0.58 to 0.66. Which statement is the correct interpretation of this interval?
    • There is a 95% probability that the true proportion falls between 0.58 and 0.66
    • 95% of residents recycle at a rate between 0.58 and 0.66
    • 95% of all possible samples will produce a sample proportion between 0.58 and 0.66
    • We are 95% confident that the true proportion of residents who recycle is between 0.58 and 0.66

    Correct answer: We are 95% confident that the true proportion of residents who recycle is between 0.58 and 0.66

    The correct interpretation is that we are 95% confident the true population proportion of residents who recycle lies between 0.58 and 0.66. The 95% refers to the long-run capture rate of the method, not a probability that this particular fixed interval contains the parameter. The interval describes the population proportion, not the behavior of individual residents or sample proportions.

71. A test of H0: p = 0.30 against Ha: p > 0.30 yields a p-value of 0.02 at a significance level of 0.05. What is the correct conclusion?
    • Reject the null hypothesis; there is convincing evidence the proportion exceeds 0.30
    • Fail to reject the null hypothesis; there is no evidence the proportion exceeds 0.30
    • Accept the null hypothesis as proven true
    • The result is inconclusive because the p-value is positive

    Correct answer: Reject the null hypothesis; there is convincing evidence the proportion exceeds 0.30

    Since the p-value of 0.02 is less than the significance level of 0.05, we reject the null hypothesis and conclude there is convincing evidence the proportion exceeds 0.30. A p-value below alpha signals a result too unlikely to attribute to chance under the null. We never accept the null as proven, and a small p-value is decisive rather than inconclusive.

72. A study reports a p-value of 0.46 when testing whether a coin is unfair. What is the best interpretation of this p-value in context?
    • There is a 46% chance the coin is fair
    • If the coin were fair, a result at least as extreme as the observed one would occur about 46% of the time, so there is no evidence the coin is unfair
    • The coin is unfair 46% of the time it is flipped
    • The probability of a Type I error in this test is 0.46

    Correct answer: If the coin were fair, a result at least as extreme as the observed one would occur about 46% of the time, so there is no evidence the coin is unfair

    The correct interpretation is that, assuming the coin is fair, a result at least as extreme as the observed one would happen about 46% of the time, which is quite common and gives no evidence the coin is unfair. A p-value is a conditional probability about the data, not the probability that the null hypothesis is true and not a fixed Type I error rate.

73. A polling firm wants to estimate the proportion of likely voters favoring a candidate but finds its margin of error is too large. Which change would reduce the margin of error?
    • Decreasing the sample size
    • Increasing the sample size
    • Raising the confidence level from 95% to 99%
    • Switching from a proportion to a count

    Correct answer: Increasing the sample size

    Increasing the sample size reduces the margin of error because the standard error of the sample proportion shrinks as the sample size grows, narrowing the interval. Decreasing the sample size would widen it, and raising the confidence level enlarges the critical value, which also widens the margin. Reporting a count instead of a proportion does not address the margin of error.

74. A medical researcher compares the proportion of patients improving under a new drug to the proportion improving under a placebo, using data from two independent groups. Which inference procedure is most appropriate?
    • A one-proportion z-interval
    • A test of a single population mean
    • A two-proportion z-test
    • A goodness-of-fit comparison of one group to a fixed value

    Correct answer: A two-proportion z-test

    A two-proportion z-test is most appropriate because the researcher is comparing categorical success proportions from two independent groups to see whether they differ. A one-proportion procedure handles only a single group, and a mean-based test applies to quantitative data rather than proportions. The setup involves two observed proportions, which is exactly what the two-proportion z-test addresses.

75. A drug company sets a very small significance level so it rarely concludes a useless drug works, but as a consequence its test often fails to detect drugs that truly do work. This tradeoff is best described by which relationship?
    • Lowering the chance of a Type I error tends to increase the chance of a Type II error and lower power
    • Lowering the chance of a Type I error also lowers the chance of a Type II error
    • Type I and Type II errors are unrelated to the significance level
    • Reducing the significance level increases the power of the test

    Correct answer: Lowering the chance of a Type I error tends to increase the chance of a Type II error and lower power

    Lowering the significance level reduces the chance of a Type I error but tends to increase the chance of a Type II error, which lowers the power to detect a real effect. Making it harder to reject the null protects against false alarms at the cost of missing true effects. The two error types trade off rather than moving together, and shrinking alpha does not raise power.

76. Before performing a one-proportion z-test, a student checks that the expected number of successes and the expected number of failures are each at least 10. Why is verifying this condition important?
    • It guarantees the sample is a simple random sample
    • It ensures the sampling distribution of the sample proportion is approximately normal so the z-procedure is valid
    • It proves the null hypothesis is true
    • It eliminates the possibility of a Type I error

    Correct answer: It ensures the sampling distribution of the sample proportion is approximately normal so the z-procedure is valid

    Checking that the expected successes and failures are each at least 10 ensures the sampling distribution of the sample proportion is approximately normal, which is what makes the z-based test valid. This large-counts condition justifies using the normal model for the proportion. It does not address whether the data were randomly sampled, prove the null, or remove the chance of a Type I error.

77. A survey records each respondent's favorite music genre. How is this variable best classified?
    • Quantitative and discrete
    • Categorical
    • Quantitative and continuous
    • A parameter

    Correct answer: Categorical

    Favorite music genre is a categorical variable because it places each respondent into a group or label rather than recording a numerical measurement. Quantitative variables, whether discrete or continuous, take number values that can be added or averaged. A parameter is a numerical summary of a population, not a type of variable.

78. Which graphical display is appropriate for showing the distribution of a single categorical variable?
    • A histogram
    • A scatterplot
    • A boxplot
    • A bar chart

    Correct answer: A bar chart

    A bar chart is appropriate for a single categorical variable because it shows the count or proportion in each category with separated bars. A histogram and a boxplot display quantitative data, and a scatterplot shows the relationship between two quantitative variables. The gaps between bars distinguish a bar chart from a histogram.

79. What key visual feature distinguishes a histogram from a bar chart?
    • A histogram uses pie slices instead of bars
    • A histogram's bars touch because the horizontal axis is a continuous numerical scale
    • A histogram always has exactly five bars
    • A histogram can only display categorical data

    Correct answer: A histogram's bars touch because the horizontal axis is a continuous numerical scale

    A histogram's bars touch because its horizontal axis represents a continuous numerical scale divided into intervals, so adjacent bins share a boundary. A bar chart leaves gaps between bars because its categories are distinct labels. Histograms display quantitative data, the opposite of the categorical-only claim.

80. A relative frequency table reports the proportion of students in each grade level. The relative frequencies in such a table should sum to what value?
    • The number of categories
    • 1
    • The total number of students
    • The largest single frequency

    Correct answer: 1

    The relative frequencies must sum to 1 because each relative frequency is a part of the whole expressed as a proportion, and the parts together make up the entire data set. If expressed as percentages they would total 100%. The raw counts, not the relative frequencies, sum to the total number of students.

81. In a stemplot (stem-and-leaf plot) of the value 47, where the stems are tens, what is the leaf?
    • 47
    • 4
    • 7
    • 0

    Correct answer: 7

    The leaf is 7 because in a stemplot the stem holds the leading digit or digits and the leaf holds the final digit. For 47 with stems representing tens, the stem is 4 and the leaf is 7. The full number 47 is not a single leaf, and 0 is not a digit of this value.

82. Which display preserves the individual data values while still showing the overall shape of a small quantitative data set?
    • A pie chart
    • A stemplot
    • A bar chart
    • A relative frequency table of categories

    Correct answer: A stemplot

    A stemplot preserves the individual data values while revealing the shape because every observation appears as a leaf attached to its stem. Histograms show shape but group values into bins, hiding the exact numbers. Pie charts and category bar charts are for categorical data and do not display quantitative values.

83. When describing the distribution of a quantitative variable, which four features should be addressed?
    • Shape, center, variability, and unusual features such as outliers
    • Color, size, label, and title
    • Mean, median, mode, and range only
    • Slope, intercept, correlation, and residuals

    Correct answer: Shape, center, variability, and unusual features such as outliers

    A complete description of a quantitative distribution addresses shape, center, variability (spread), and unusual features such as outliers or gaps. Reporting only the mean, median, mode, and range omits shape and unusual features. Slope, intercept, and correlation describe relationships between two variables, not a single distribution.

84. A dotplot of quiz scores shows two clearly separated clusters of dots with a gap between them. What does this most directly indicate about the distribution's shape?
    • It is uniform
    • It is roughly symmetric and bell-shaped
    • It appears bimodal
    • It has no center

    Correct answer: It appears bimodal

    The distribution appears bimodal because two distinct clusters of dots correspond to two peaks in the data. A uniform distribution would show roughly equal heights across the range, and a bell-shaped distribution would have a single central peak. Every distribution has a center, so that option is incorrect.

85. The 30th percentile of a distribution of incomes is $42,000. What does this value mean?
    • About 30% of incomes are at or below $42,000
    • Exactly 30 people earn $42,000
    • About 30% of incomes are above $42,000
    • The average income is $42,000

    Correct answer: About 30% of incomes are at or below $42,000

    It means about 30% of the incomes are at or below $42,000, because a percentile gives the percentage of observations falling at or below a given value. It is not a count of people, and it is the lower 30%, not the upper portion. A percentile describes position, not the average.

86. A value at the 90th percentile of a test-score distribution can best be described as which of the following?
    • A value that 90% of scores exceed
    • A value at or below which about 90% of scores fall
    • A value equal to 90 points
    • A value exactly nine standard deviations above the mean

    Correct answer: A value at or below which about 90% of scores fall

    The 90th percentile is the value at or below which about 90% of the scores fall, marking a relatively high position in the distribution. It is not a value most scores exceed, nor is it necessarily 90 points. Percentiles measure relative standing, not standard deviations.

87. A z-score is computed for a data value and turns out to be negative. What does this indicate?
    • The value is below the mean
    • The value is an outlier
    • The value is above the mean
    • A calculation error was made because z-scores cannot be negative

    Correct answer: The value is below the mean

    A negative z-score indicates the value lies below the mean, since the z-score is the value minus the mean divided by the standard deviation. Values below the mean produce a negative numerator. Z-scores can certainly be negative, and a negative z-score does not by itself mark an outlier.

88. Two students take different exams. One has a z-score of 1.2 on her exam and the other has a z-score of 0.8 on his. Why does comparing z-scores allow a fair comparison?
    • Z-scores convert each value to a common scale of standard deviations from its own mean
    • Z-scores convert all scores to a 100-point scale
    • Z-scores ignore the standard deviation
    • Z-scores can only be compared if the exams are identical

    Correct answer: Z-scores convert each value to a common scale of standard deviations from its own mean

    Comparing z-scores is fair because each z-score expresses a value as the number of standard deviations from its own distribution's mean, standardizing different scales. The student with a z-score of 1.2 stands higher relative to her group. Z-scores depend on the standard deviation and do not require identical exams or a 100-point scale.

89. A density curve is used to model a distribution. What must be true of the total area under any density curve?
    • It equals the number of data values
    • It equals the mean
    • It can be any positive number
    • It equals 1

    Correct answer: It equals 1

    The total area under any density curve equals 1 because the curve represents the entire distribution as proportions, and all proportions together make up the whole. Areas under regions of the curve correspond to proportions of data. The area is fixed at 1 regardless of the number of data values or the mean.

90. On a density curve that is skewed to the right, how do the locations of the mean and median compare?
    • The median lies to the right of the mean
    • The mean lies to the right of the median
    • They are always at the exact same point
    • The mean lies at the peak of the curve

    Correct answer: The mean lies to the right of the median

    On a right-skewed density curve the mean lies to the right of the median because the long right tail pulls the mean toward the larger values while the median stays near the bulk of the area. The mean and median coincide only on a symmetric curve. The mean is not generally located at the peak.

91. In a perfectly symmetric, single-peaked density curve, where are the mean and median located?
    • The mean is left of the median
    • Both lie at the center of the curve
    • The median is at the peak but the mean is in the tail
    • Both lie at the far right of the curve

    Correct answer: Both lie at the center of the curve

    In a symmetric, single-peaked density curve both the mean and the median lie at the center, which is also the peak, because the curve balances evenly on either side. Skew is what separates the mean from the median. Neither measure is pulled into a tail when the curve is symmetric.

92. Approximately what percentage of observations in a normal distribution fall within one standard deviation of the mean according to the empirical rule?
    • About 50%
    • About 99.7%
    • About 68%
    • About 95%

    Correct answer: About 68%

    About 68% of observations fall within one standard deviation of the mean under the empirical rule. About 95% lie within two standard deviations and about 99.7% within three. The 50% figure corresponds to the median splitting the distribution in half, not a one-standard-deviation interval.

93. Battery lifetimes are approximately normal with a mean of 40 hours and a standard deviation of 5 hours. Within which interval do about 99.7% of the lifetimes fall?
    • 35 to 45 hours
    • 25 to 55 hours
    • 30 to 50 hours
    • 20 to 60 hours

    Correct answer: 25 to 55 hours

    About 99.7% of lifetimes fall between 25 and 55 hours, found by taking the mean plus or minus three standard deviations: 40 minus 3 times 5 is 25, and 40 plus 3 times 5 is 55. The empirical rule assigns 99.7% to three standard deviations. The interval 30 to 50 covers two standard deviations (about 95%).

94. Which measure of variability uses every value in the data set in its calculation?
    • The range
    • The interquartile range
    • The standard deviation
    • The median

    Correct answer: The standard deviation

    The standard deviation uses every value in the data set because it is based on each observation's squared deviation from the mean. The range uses only the maximum and minimum, the interquartile range uses only the quartiles, and the median is a measure of center, not variability.

95. A data set of house ages has a minimum of 2 and a maximum of 88. What is the range?
    • 88
    • 45
    • 86
    • 90

    Correct answer: 86

    The range is 86, found by subtracting the minimum from the maximum: 88 minus 2 equals 86. The range measures the total spread between the extreme values. Using the maximum alone (88) or adding the values (90) does not give the range.

96. Why is the standard deviation, rather than the variance, often preferred when describing the spread of a single quantitative variable?
    • The variance is always negative
    • The standard deviation ignores outliers entirely
    • The variance cannot be calculated for real data
    • The standard deviation is in the same units as the original data

    Correct answer: The standard deviation is in the same units as the original data

    The standard deviation is often preferred because it is expressed in the same units as the original data, making it easier to interpret, while the variance is in squared units. The variance is never negative and can certainly be computed. Neither measure ignores outliers; both are affected by extreme values.

97. If every value in a data set is multiplied by 3, what happens to the median?
    • It stays the same
    • It increases by 3
    • It is divided by 3
    • It is multiplied by 3

    Correct answer: It is multiplied by 3

    The median is multiplied by 3 because multiplying every value by a constant scales all measures of center and spread by that same constant. The middle value, like all others, becomes three times larger. Adding a constant would instead shift the median, but here the values are multiplied.

98. Adding a constant of 5 to every value in a data set has which effect on the measures of the distribution?
    • Both the mean and the standard deviation increase by 5
    • The mean increases by 5 but the standard deviation is unchanged
    • The mean is unchanged but the standard deviation increases by 5
    • Neither the mean nor the standard deviation changes

    Correct answer: The mean increases by 5 but the standard deviation is unchanged

    Adding 5 to every value increases the mean by 5 while leaving the standard deviation unchanged, because a uniform shift moves the center but does not alter the distances between values. Measures of center respond to added constants, but measures of spread such as standard deviation do not.

99. A frequency table shows 40 freshmen, 30 sophomores, 20 juniors, and 10 seniors. What is the relative frequency of juniors?
    • 0.20
    • 0.10
    • 0.30
    • 20

    Correct answer: 0.20

    The relative frequency of juniors is 0.20, found by dividing the 20 juniors by the total of 100 students. Relative frequency expresses a category's count as a proportion of the whole. The value 20 is the raw count, not a proportion, and 0.10 corresponds to seniors.

100. When is the median generally a more appropriate measure of center than the mean?
     • When the distribution is strongly skewed or has outliers
     • When the distribution is perfectly symmetric
     • When every value is identical
     • When the data are categorical

     Correct answer: When the distribution is strongly skewed or has outliers

     The median is generally more appropriate when the distribution is strongly skewed or contains outliers, because it resists the pull of extreme values that distort the mean. In a symmetric distribution the mean and median agree, so either works. Measures of center do not apply to categorical data.

101. A cumulative relative frequency graph (ogive) reaches a height of 0.75 at the value x = 80. What does this tell you?
     • Exactly 75 values equal 80
     • About 75% of the data values are above 80
     • The value 80 is the mean of the data
     • About 75% of the data values are at or below 80

     Correct answer: About 75% of the data values are at or below 80

     It tells you that about 75% of the data values are at or below 80, because a cumulative relative frequency graph plots the running proportion of data up to each value. The height of 0.75 marks 80 as roughly the 75th percentile, or third quartile. It does not represent a count or the mean.

102. Which of the following is a quantitative variable?
     • Eye color of survey respondents
     • Number of text messages sent per day
     • Type of pet owned
     • Preferred brand of cereal

     Correct answer: Number of text messages sent per day

     The number of text messages sent per day is quantitative because it is a numerical count that can be averaged and ordered meaningfully. Eye color, type of pet, and preferred cereal brand are all categorical, since they sort respondents into groups rather than measuring a quantity.

103. A histogram of reaction times is described as right-skewed. Which statement about its tail is correct?
     • The tail extends toward the larger values on the right
     • The tail extends toward the smaller values on the left
     • There is no tail in a skewed distribution
     • The tail is always exactly symmetric

     Correct answer: The tail extends toward the larger values on the right

     In a right-skewed histogram the tail extends toward the larger values on the right, which is where the distribution is named for. Most observations cluster on the lower end with a few high values stretching the tail. A left-skewed distribution would have its tail toward the smaller values.

104. A pie chart and a bar chart both summarize the same categorical variable. What is one advantage a bar chart has over a pie chart?
     • A bar chart can display quantitative intervals
     • A bar chart makes it easier to compare the sizes of categories directly
     • A bar chart shows the exact individual data values
     • A bar chart requires the categories to sum to 100%

     Correct answer: A bar chart makes it easier to compare the sizes of categories directly

     A bar chart makes it easier to compare category sizes directly because the heights of bars are simpler to judge than the angles of pie slices. Both displays are for categorical data and do not show individual quantitative values. Neither requires the categories to total 100%, though relative frequencies do.

105. A data set of ages includes the values 5, 6, 6, 7, 8, and 95. Which measure of center would be most misleading for describing a typical age, and why?
     • The median, because it ignores the value 95
     • The mean, because the value 95 inflates it far above the typical age
     • The mode, because it equals 6
     • The minimum, because it equals 5

     Correct answer: The mean, because the value 95 inflates it far above the typical age

     The mean is most misleading here because the extreme value 95 pulls it far above the cluster of younger ages, so it no longer represents a typical age. The median resists that single outlier and better reflects the center. The minimum and mode are not measures of typical center in this context.

106. A standardized assessment is normally distributed. A score with a z-score of 0 corresponds to which value?
     • The minimum score
     • The mean score
     • The maximum score
     • A score one standard deviation above average

     Correct answer: The mean score

     A z-score of 0 corresponds to the mean score, because the z-score formula subtracts the mean and a result of zero occurs only when the value equals the mean. It marks the center of the distribution, not an extreme. A z-score of 1, not 0, would be one standard deviation above the mean.

107. A distribution of values is described as roughly uniform. Which dotplot or histogram pattern matches this description?
     • A single tall peak in the center that tapers off on both sides
     • One long tail stretching to the right
     • Bars of approximately equal height across the range of values
     • Two separate clusters with a gap between them

     Correct answer: Bars of approximately equal height across the range of values

     A roughly uniform distribution shows bars of approximately equal height across the range of values, meaning each interval contains about the same number of observations. A single central peak describes a bell shape, one long tail describes a skewed shape, and two clusters describe a bimodal shape.

108. A manager arranges all 800 employees in a list ordered by hire date, randomly picks one of the first 10 names, and then selects every 10th employee after that. Which sampling method is this?
     • Systematic random sampling
     • Stratified random sampling
     • Cluster sampling
     • Simple random sampling

     Correct answer: Systematic random sampling

     This is systematic random sampling because a random starting point is chosen and then every kth member of an ordered list is selected. Choosing every 10th employee after a random start defines the systematic pattern. It is not an SRS, since not every group of employees is equally likely, and it does not form strata or sample whole clusters.

109. A researcher wants to estimate the average commute time of city workers but only has a list of downtown office buildings, not a list of all workers. What is the main reason this list could cause undercoverage?
     • The list will make the sample too large to handle
     • Workers outside downtown offices have no chance of being selected
     • Random assignment cannot be applied to the list
     • It forces the study to become an experiment

     Correct answer: Workers outside downtown offices have no chance of being selected

     The answer is that workers outside downtown offices have no chance of being selected, which is the definition of undercoverage. A sampling frame that omits part of the population leaves those members unreachable, biasing the estimate. The flaw is the incomplete frame, not sample size, randomization, or study type.

110. In a phone survey, interviewers can only reach people who answer their landline during business hours. People who work during those hours and never pick up are systematically excluded. This is an example of what type of error?
     • Sampling variability
     • Confounding
     • Nonresponse bias
     • Voluntary response bias

     Correct answer: Nonresponse bias

     This describes nonresponse bias, since selected individuals who cannot be reached or do not answer differ systematically from those who do. The working people excluded may have different commute or schedule patterns, skewing results. It is not random sampling variability, and no one is volunteering or being assigned a treatment here.

111. A pollster asks, 'Don't you agree that the wasteful new tax should be repealed?' Why is this question problematic?
     • It is too short to gather useful information
     • It requires a random sample to be valid
     • The wording is leading and can bias responses toward one answer
     • It introduces a confounding variable into the survey

     Correct answer: The wording is leading and can bias responses toward one answer

     The answer is that the wording is leading and can bias responses, because describing the tax as 'wasteful' pushes respondents toward agreeing it should be repealed. Loaded language is a recognized source of response bias in surveys. The issue is question phrasing, not length, sampling method, or confounding, which applies to experiments.

112. An agricultural scientist divides a field into four plots, applies the same fertilizer to all four, and measures yield. A colleague notes the study cannot detect whether the fertilizer helps. What essential element of an experiment is missing?
     • A response variable
     • Random sampling of plots
     • A larger field
     • A comparison group receiving a different or no treatment

     Correct answer: A comparison group receiving a different or no treatment

     The missing element is a comparison group receiving a different or no treatment, since without one there is no baseline to judge the fertilizer's effect. Comparison is a core principle of experimental design. A response variable (yield) is present, and random sampling and field size do not address the lack of comparison.

113. In an experiment, subjects are assigned to treatments, and neither the subjects nor the people measuring outcomes know who received which treatment. This design feature is called what?
     • Double-blind
     • Stratification
     • Replication
     • Blocking

     Correct answer: Double-blind

     The answer is double-blind, because both the subjects and those assessing the response are unaware of the treatment assignments. This guards against bias from expectations on either side. It differs from stratification and blocking, which group subjects, and from replication, which refers to using many experimental units.

114. A drug study gives one group the new medication and another group a pill with no active ingredient. Several patients in the no-medication group report feeling better anyway. This improvement among those receiving the inactive pill is best described as what?
     • A confounding variable
     • Sampling bias
     • The placebo effect
     • Replication

     Correct answer: The placebo effect

     This is the placebo effect, the phenomenon where subjects respond favorably simply because they believe they are being treated. It is exactly why studies include a placebo control. It is not confounding, not a sampling flaw, and not replication, which refers to applying treatments to many units.

115. An exercise study expects that men and women may respond differently to a training plan. Researchers separate subjects into a male block and a female block, then randomly assign treatments within each block. What is the purpose of this blocking?
     • To allow the results to generalize to the whole country
     • To turn the study into an observational study
     • To guarantee no placebo effect occurs
     • To reduce variability by accounting for a known source of differences

     Correct answer: To reduce variability by accounting for a known source of differences

     The purpose of blocking is to reduce variability by accounting for a known source of differences, here sex. Comparing treatments within similar blocks isolates the treatment effect from sex-related variation. Blocking does not control generalization, change the study to observational, or address the placebo effect.

116. A researcher conducts a matched-pairs experiment in which each subject receives both treatments in a random order at different times. What is the main advantage of this matched-pairs design?
     • Each subject serves as their own control, reducing person-to-person variability
     • It eliminates the need for any randomization
     • It makes the study an observational study
     • It guarantees the sample represents the population

     Correct answer: Each subject serves as their own control, reducing person-to-person variability

     The advantage is that each subject serves as their own control, reducing person-to-person variability so differences are attributed to the treatments. Randomizing the order of treatments is still required. The design remains an experiment and does not govern how well the sample represents the population.

117. A study of 60 plants applies a growth hormone to 30 plants and none to the other 30. Why is applying the treatment to 30 plants rather than just 1 important?
     • It makes the experiment double-blind
     • It removes the need for a control group
     • Replication across many units allows real effects to be distinguished from chance variation
     • It converts the experiment into a census

     Correct answer: Replication across many units allows real effects to be distinguished from chance variation

     The answer is that replication across many units lets researchers separate a genuine treatment effect from random plant-to-plant variation. A single plant could differ for unrelated reasons. Using many units does not create blinding, eliminate the control group, or make the study a census.

118. A polling firm correctly takes a simple random sample of 1,000 adults but discovers afterward that wealthy respondents were far more willing to complete the long survey than others. The conclusions could still be biased mainly because of what?
     • The sample size was too small
     • Random assignment was not used
     • Nonresponse that differs by income group
     • The use of stratified sampling

     Correct answer: Nonresponse that differs by income group

     The bias stems from nonresponse that differs by income group, since wealthier respondents completed the survey at higher rates, skewing results. A good random selection cannot fix who chooses not to respond. The problem is not sample size, random assignment, or stratification, which was not used.

119. Which of the following best describes the difference between the population and a sample in a statistical study?
     • The population is always smaller than the sample
     • The sample includes everyone, while the population is only those who respond
     • They are identical in every well-designed study
     • The population is the entire group of interest, while the sample is the subset actually examined

     Correct answer: The population is the entire group of interest, while the sample is the subset actually examined

     The answer is that the population is the entire group of interest while the sample is the subset actually examined. Researchers study the sample to draw conclusions about the larger population. The population is not smaller than the sample, the sample is not everyone, and the two are not identical.

120. A survey on personal income asks respondents to state their salary to a live interviewer face to face. Many people overstate their income. This tendency to give answers seen as more favorable is best classified as what?
     • Response bias
     • Undercoverage
     • Sampling variability
     • Confounding

     Correct answer: Response bias

     This is response bias, where respondents give inaccurate answers, here inflating income to appear more favorable. The face-to-face setting encourages socially desirable responses. It is not undercoverage, which concerns excluded groups, nor sampling variability or confounding, which belong to experiments.

121. A scientist wants to know if a tutoring program improves test scores and randomly assigns students to receive tutoring or not. Students who knew they were being tutored tried harder simply because of the attention. To reduce this, the tutoring and control activities should be made as similar as possible. The unwanted effect being reduced is best described as what?
     • Undercoverage of the student population
     • A lurking effect from subjects' awareness of treatment, similar to a placebo effect
     • Sampling bias in the selection of students
     • Lack of replication in the experiment

     Correct answer: A lurking effect from subjects' awareness of treatment, similar to a placebo effect

     The answer is a lurking effect from subjects' awareness, similar to a placebo effect, where extra effort comes from knowing one is in the special group rather than from tutoring itself. Making activities comparable controls for it. This is not undercoverage, sampling bias, or a replication issue.

122. Researchers randomly select 50 schools from a district and then survey all teachers within those selected schools. This combination is best described as what?
     • A simple random sample of teachers
     • Stratified sampling, with schools as strata
     • A systematic sample of teachers
     • Cluster sampling, with schools as clusters

     Correct answer: Cluster sampling, with schools as clusters

     The answer is cluster sampling with schools as clusters, because whole schools are randomly chosen and every teacher within them is surveyed. Sampling entire selected groups in full is the hallmark of cluster sampling. It is not an SRS or systematic sample of individual teachers, nor stratified sampling, which samples within every group.

123. An experiment finds a real, statistically convincing difference between two treatments, but all subjects were unpaid volunteers who responded to an online ad. What limitation does this place on the conclusion?
     • No cause-and-effect conclusion can be drawn at all
     • Random assignment was clearly not used
     • The results may not generalize beyond people like the volunteers
     • The study must be an observational study

     Correct answer: The results may not generalize beyond people like the volunteers

     The limitation is that results may not generalize beyond people like the volunteers, since they were not a random sample of any broader population. Random assignment still supports a causal claim for these subjects, but generalization requires random selection. The study is a valid experiment, just limited in scope of inference.

124. A study randomly selects participants from the entire population and randomly assigns each to one of two diets. Which two conclusions are most appropriately supported by this design?
     • Causation, and generalization to the population
     • Only association, and no generalization
     • Generalization only, but no causal claim
     • Neither causation nor generalization

     Correct answer: Causation, and generalization to the population

     The answer is causation and generalization to the population. Random assignment supports a cause-and-effect conclusion, while random selection from the population supports generalizing the result. Having both random selection and random assignment is what permits both inferences at once.

125. A factory inspector wants every item in a shipment to have a known, equal chance of selection and uses a random number generator to pick item ID numbers. If two items happen to share the exact same ID due to a labeling error, what problem does this create for the sampling process?
     • It makes the study an experiment
     • It introduces the placebo effect
     • It guarantees nonresponse bias
     • It creates undercoverage because identical IDs cannot both be distinguished and selected fairly

     Correct answer: It creates undercoverage because identical IDs cannot both be distinguished and selected fairly

     The answer is that it creates undercoverage, since two items sharing one ID cannot be told apart, so at least one lacks a fair, distinct chance of selection. A proper frame requires every unit to be uniquely identifiable. This is not an experiment, a placebo issue, or nonresponse.

126. A health magazine reports that readers who eat breakfast tend to weigh less, based on a survey of its subscribers, and concludes that eating breakfast causes weight loss. What is the most accurate criticism?
     • The survey used too large a sample
     • Random assignment was used incorrectly
     • This is observational data, so a lurking variable could explain the link
     • Blocking should have been applied to the subscribers

     Correct answer: This is observational data, so a lurking variable could explain the link

     The accurate criticism is that this is observational data, so a lurking variable such as overall health habits could explain the association. Without random assignment of who eats breakfast, causation cannot be claimed. The issue is not sample size, and random assignment and blocking were never part of a survey.

127. In designing a survey, why is it generally better to use a chance-based method to select respondents rather than letting the researcher choose who seems representative?
     • Chance selection always produces a larger sample
     • Chance selection guarantees zero sampling error
     • Chance selection allows a causal conclusion
     • Chance selection avoids the researcher's conscious or unconscious bias in choosing

     Correct answer: Chance selection avoids the researcher's conscious or unconscious bias in choosing

     The answer is that chance selection avoids the researcher's conscious or unconscious bias in choosing who participates. Letting a person pick 'representative' individuals invites systematic favoring of certain types. Chance methods do not guarantee larger samples, eliminate sampling error, or, in a survey, enable causal claims.

128. A treatment group and a control group in a well-run experiment differ in their average response. Before concluding the treatment caused the difference, what must researchers also consider?
     • Whether the difference is larger than what random assignment alone could plausibly produce
     • Whether the sample was a census of the population
     • Whether voluntary response was used
     • Whether the population was divided into strata

     Correct answer: Whether the difference is larger than what random assignment alone could plausibly produce

     The answer is whether the difference is larger than random assignment alone could plausibly produce, since chance creates some difference even with no real effect. Only a difference too large to be explained by chance signals a treatment effect. Census, voluntary response, and strata are unrelated to this judgment.

129. A company emails a feedback link to all customers and analyzes only the responses it receives. Compared with this approach, why would randomly selecting a subset of customers and actively contacting them give more trustworthy results?
     • It would produce a census of all customers
     • It would turn the study into an experiment
     • It avoids voluntary response bias by not relying on who chooses to reply
     • It removes the need for a sampling frame

     Correct answer: It avoids voluntary response bias by not relying on who chooses to reply

     The answer is that random selection with active follow-up avoids voluntary response bias, which arises when only self-motivated customers reply. Reaching a chosen subset reduces the dominance of strong opinions. It does not create a census, become an experiment, or remove the need for a sampling frame.

130. An experimenter gives every subject the new energy drink and then asks if they feel more alert; most say yes. Why can the experimenter not conclude the drink increases alertness?
     • The sample was selected without replacement
     • Stratified sampling was not used
     • There is no comparison group, so the placebo effect and other causes cannot be ruled out
     • The response variable was measured incorrectly

     Correct answer: There is no comparison group, so the placebo effect and other causes cannot be ruled out

     The answer is that with no comparison group, the placebo effect and other explanations cannot be ruled out, so the reported alertness may not come from the drink. A control group is needed to isolate the effect. The problem is the missing comparison, not replacement, stratification, or measurement of the response.

131. A researcher selects a sample by writing each member of the population on an identical slip, mixing the slips thoroughly in a bowl, and drawing slips without looking. Which sampling method does this physical procedure carry out?
     • Simple random sampling
     • Stratified random sampling
     • Cluster sampling
     • Convenience sampling

     Correct answer: Simple random sampling

     The answer is simple random sampling, because identical slips drawn blindly give every possible group of members an equal chance of selection. The bowl-and-slips method is a classic way to physically implement an SRS. It does not form strata, sample whole clusters, or pick whoever is easiest, which would be convenience sampling.

132. A spinner is divided into regions so that the probability of landing on red is 0.25, on blue is 0.35, and on green is some unknown value, with no other outcomes possible. What must the probability of green be?
     • 0.60
     • 0.30
     • 0.40
     • 0.50

     Correct answer: 0.40

     The probability of green must be 0.40 because the probabilities of all possible outcomes in a sample space must sum to 1: 1 minus 0.25 minus 0.35 equals 0.40. Since red, blue, and green are the only outcomes, their probabilities are exhausted by this total. Values such as 0.30 or 0.60 would make the probabilities fail to add to 1.

133. For any event A in a sample space, what is the relationship between the probability of A and the probability of its complement, not A?
     • They are always equal to each other
     • They sum to 1
     • They multiply to 1
     • Their difference is always 0.5

     Correct answer: They sum to 1

     The probability of A and the probability of its complement sum to 1 because every outcome either is in A or is not in A, covering the entire sample space exactly once. This is the complement rule, written as P(not A) equals 1 minus P(A). The two probabilities are not generally equal, and they do not multiply to 1.

134. Events A and B are not mutually exclusive. According to the general addition rule, how is the probability that A or B occurs calculated?
     • P(A) plus P(B)
     • P(A) plus P(B) minus P(A and B)
     • P(A) times P(B)
     • P(A) minus P(B)

     Correct answer: P(A) plus P(B) minus P(A and B)

     The probability of A or B equals P(A) plus P(B) minus P(A and B) because simply adding the two probabilities counts the overlap region twice, so the joint probability must be subtracted once. This general addition rule applies whether or not the events overlap. Just adding P(A) and P(B) is correct only when the events are mutually exclusive.

135. A bag contains 4 red and 6 blue marbles. A marble is drawn, its color recorded, and it is replaced before a second draw. What is the probability both marbles drawn are red?
     • 0.40
     • 0.12
     • 0.08
     • 0.16

     Correct answer: 0.16

     The probability both are red is 0.16 because drawing with replacement keeps the trials independent, so multiply the probabilities: 0.4 times 0.4 equals 0.16. Replacement restores the bag to 4 red out of 10 for the second draw. Failing to multiply, or treating the draws as dependent, would give a different value.

136. A bag contains 4 red and 6 blue marbles. Two marbles are drawn without replacement. What is the probability that both are red?
     • 0.1333
     • 0.16
     • 0.40
     • 0.20

     Correct answer: 0.1333

     The probability both are red is about 0.1333 because without replacement the second draw depends on the first: $\frac{4}{10}$ times $\frac{3}{9}$ equals $\frac{12}{90}$, which is approximately 0.1333. After removing one red marble, only 3 red remain among 9 marbles. Using 0.16 would wrongly treat the draws as independent with replacement.

137. The probability distribution of a discrete random variable lists each possible value with its probability. Which condition must this list of probabilities satisfy?
     • Each probability must be greater than 0.5
     • The probabilities must all be equal
     • Each probability is between 0 and 1, and they sum to 1
     • The probabilities must sum to the number of outcomes

     Correct answer: Each probability is between 0 and 1, and they sum to 1

     A valid probability distribution requires each probability to be between 0 and 1 and the probabilities to sum to 1, since these are the basic rules every probability must obey. The values need not be equal, and there is no requirement that any single probability exceed 0.5. Summing to the number of outcomes would violate the rule that total probability equals 1.

138. A discrete random variable X takes the value 0 with probability 0.5, the value 1 with probability 0.3, and the value 2 with probability 0.2. What is the expected value of X?
     • 1.0
     • 1.5
     • 0.5
     • 0.7

     Correct answer: 0.7

     The expected value is 0.7, computed as the sum of each value times its probability: 0 times 0.5 plus 1 times 0.3 plus 2 times 0.2 equals 0 plus 0.3 plus 0.4, which is 0.7. Expected value weights each outcome by its probability rather than averaging the values evenly. The simple average of 0, 1, and 2 would mistakenly give 1.0.

139. When two events are independent, which statement about conditional probability is true?
     • The probability of A given B equals zero
     • The probability of A given B equals the probability of B
     • The probability of A given B equals the probability of A
     • The probability of A given B is always larger than the probability of A

     Correct answer: The probability of A given B equals the probability of A

     For independent events, the probability of A given B equals the probability of A, because knowing that B occurred provides no information that changes A's probability. This is the defining feature of independence. A conditional probability of zero would describe mutually exclusive events, not independent ones.

140. A random variable that can take any value within an interval, such as the exact height of a randomly chosen adult, is best described as which type of variable?
     • A discrete random variable
     • A continuous random variable
     • A categorical variable
     • A binomial random variable

     Correct answer: A continuous random variable

     Exact height is a continuous random variable because it can take any value within an interval rather than only separated, countable values. Discrete random variables, by contrast, take a countable set of values such as whole-number counts. Height is numeric rather than categorical, and it is not restricted to a binomial count of successes.

141. Let X be a random variable with mean 10. A new variable is defined as Y equals 3 times X plus 5. What is the mean of Y?
     • 15
     • 30
     • 35
     • 18

     Correct answer: 35

     The mean of Y is 35 because the linear transformation rule gives the mean of 3X plus 5 as 3 times the mean of X plus 5: 3 times 10 plus 5 equals 30 plus 5, which is 35. Both the multiplier and the added constant affect the mean. Forgetting to add the constant 5 would wrongly yield 30.

142. Let X be a random variable with standard deviation 4. A new variable is defined as Y equals 2 times X plus 7. What is the standard deviation of Y?
     • 8
     • 15
     • 11
     • 4

     Correct answer: 8

     The standard deviation of Y is 8 because multiplying a random variable by a constant multiplies the standard deviation by the absolute value of that constant, while adding a constant does not change spread: 2 times 4 equals 8. The added 7 shifts all values equally and leaves variability unchanged. Adding 7 to the standard deviation would be an error.

143. Two independent random variables have means of 12 and 5 and variances of 9 and 16, respectively. What is the variance of their sum?
     • 5
     • 625
     • 25
     • 12.5

     Correct answer: 25

     The variance of the sum is 25 because for independent random variables variances add: 9 plus 16 equals 25. This addition of variances holds for both sums and differences of independent variables. Standard deviations, unlike variances, cannot simply be added, so combining them directly would be incorrect.

144. Two independent random variables have means 12 and 5 and variances 9 and 16. What is the variance of their difference, the first minus the second?
     • 7
     • 25
     • Minus 7
     • 5

     Correct answer: 25

     The variance of the difference is 25 because variances of independent random variables add even when the variables are subtracted: 9 plus 16 equals 25. Subtraction does not subtract variances, since variability accumulates regardless of the sign. Reporting 7 by subtracting the variances would be a common mistake.

145. In a binomial setting with 20 trials and probability of success 0.3 on each trial, what is the standard deviation of the number of successes?
     • 6
     • 4.2
     • About 1.45
     • About 2.05

     Correct answer: About 2.05

     The standard deviation is about 2.05 because the binomial standard deviation is $\sqrt{n p (1 - p)}$: $\sqrt{20 \times 0.3 \times 0.7}$, which is $\sqrt{4.2}$, approximately 2.05. The value 4.2 is the variance, not the standard deviation. The mean, 6, is np and is unrelated to spread.

146. A fair coin is flipped repeatedly until the first head appears. On average, how many flips are expected before the first head, given a success probability of 0.5 on each flip?
     • 1
     • 2
     • 0.5
     • 4

     Correct answer: 2

     The expected number of flips is 2 because the mean of a geometric random variable is 1 divided by the success probability: 1 divided by 0.5 equals 2. On average, it takes the reciprocal of the per-trial success probability to reach the first success. Using 0.5 itself confuses the probability with the expected waiting time.

147. A survey finds that 70% of households own a pet, and among pet-owning households 40% own a dog. What is the probability that a randomly chosen household both owns a pet and owns a dog?
     • 0.40
     • 0.70
     • 0.28
     • 1.10

     Correct answer: 0.28

     The probability is 0.28 because the general multiplication rule multiplies the probability of the first event by the conditional probability of the second given the first: 0.70 times 0.40 equals 0.28. The 40% applies only within the pet-owning group, so it must be scaled by the 70%. Adding the percentages would incorrectly exceed both individual values.

148. A study reports P(A) = 0.6, P(B) = 0.5, and P(A and B) = 0.2. What is the probability that A or B occurs?
     • 0.9
     • 1.1
     • 0.7
     • 0.3

     Correct answer: 0.9

     The probability of A or B is 0.9 because the general addition rule subtracts the overlap once: 0.6 plus 0.5 minus 0.2 equals 0.9. Without subtracting the joint probability, the overlap would be double-counted, giving an impossible value above 1. The result stays within the valid 0-to-1 range only after the subtraction.

149. A simulation is used to estimate the probability that at least two people in a group of 30 share a birthday. Why might a simulation be a reasonable approach for this problem?
     • Because the exact theoretical probability is impossible to define
     • Because simulation can approximate a probability that is tedious to compute exactly
     • Because simulation always gives the exact answer
     • Because probability rules do not apply to birthdays

     Correct answer: Because simulation can approximate a probability that is tedious to compute exactly

     Simulation is reasonable because it can approximate a probability that is tedious to compute exactly, repeatedly modeling the random process and recording the proportion of times the event occurs. The theoretical probability does exist and follows probability rules, but the calculation is laborious. Simulations give estimates that improve with more trials, not guaranteed exact answers.

150. A teacher claims that whether a student passed an exam is independent of whether the student attended a review session. Which equality, if true, would confirm this independence?
     • P(passed and attended) equals zero
     • P(passed given attended) equals P(passed)
     • P(passed) equals P(attended)
     • P(passed or attended) equals 1

     Correct answer: P(passed given attended) equals P(passed)

     Independence is confirmed if P(passed given attended) equals P(passed), because that shows attending the review session does not change the probability of passing. Equal marginal probabilities or a joint probability of zero do not establish independence; a zero joint probability would instead indicate mutually exclusive events. Comparing the conditional probability to the unconditional one is the correct test.

151. A discrete random variable representing the number of cars passing a checkpoint in one minute is most naturally measured on what kind of scale?
     • Any real number in an interval
     • Categories with no order
     • Whole-number counts
     • Percentages between 0 and 100

     Correct answer: Whole-number counts

     The number of cars is measured as whole-number counts because you can have 0, 1, 2, or more cars but never a fraction of a car, which is the hallmark of a discrete random variable. Continuous variables, by contrast, take any real value in an interval. Counts are numeric and ordered, not unordered categories or forced percentages.

152. Two fair dice are rolled. What is the probability that the sum of the two dice equals 7?
     • $\frac{1}{6}$
     • $\frac{1}{12}$
     • $\frac{1}{9}$
     • $\frac{7}{36}$

     Correct answer: $\frac{1}{6}$

     The probability the sum is 7 is $\frac{1}{6}$ because 6 of the 36 equally likely outcomes produce a sum of 7, and 6 divided by 36 equals $\frac{1}{6}$. The favorable pairs are (1,6), (2,5), (3,4), (4,3), (5,2), and (6,1). Counting fewer favorable outcomes, such as treating order as irrelevant, would understate the probability.

153. A weighted die is constructed so that the probability of rolling a 6 is 0.4 and the probabilities of rolling 1 through 5 are equal. What is the probability of rolling a 3?
     • 0.20
     • 0.10
     • 0.40
     • 0.12

     Correct answer: 0.12

     The probability of rolling a 3 is 0.12 because the remaining 0.6 of total probability, after the 0.4 for a 6, is split equally among the five other faces: 0.6 divided by 5 equals 0.12. Total probability across all faces must equal 1. Assuming a fair value of about 0.167 would ignore the weighting of the die.

154. A binomial random variable counts successes in 50 independent trials with success probability 0.6. What is the mean number of successes?
     • 30
     • 20
     • 12
     • 25

     Correct answer: 30

     The mean number of successes is 30 because the binomial mean equals the number of trials times the success probability: 50 times 0.6 equals 30. This expected count reflects the long-run average successes across many repetitions of the 50 trials. Using the failure probability of 0.4 would wrongly give 20.

155. In a game, a player wins $5 with probability 0.2 and loses $2 with probability 0.8. Over many plays, what is the expected gain or loss per play?
     • A loss of $0.60
     • A gain of $0.60
     • A gain of $3.00
     • A loss of $2.00

     Correct answer: A loss of $0.60

     The expected result is a loss of $0.60 per play because expected value sums each outcome times its probability: 5 times 0.2 plus (negative 2) times 0.8 equals 1 minus 1.6, which is negative 0.60. A negative expected value indicates a long-run average loss. Ignoring the loss term, or its sign, would misstate the result as a gain.

156. What is the difference between a parameter and a statistic in the context of sampling distributions?
     • A parameter describes a sample, while a statistic describes a population
     • A parameter describes a population, while a statistic describes a sample
     • A parameter and a statistic are two names for the same quantity
     • A parameter is always larger in value than the corresponding statistic

     Correct answer: A parameter describes a population, while a statistic describes a sample

     A parameter describes a population, while a statistic describes a sample. A parameter is a fixed but usually unknown numerical summary of the whole population, and a statistic is computed from sample data to estimate it. The roles are not reversed, the two terms are not interchangeable, and there is no rule forcing a parameter to be larger than a statistic.

157. An estimator is described as biased. What does bias refer to in a sampling distribution?
     • How spread out the values of the statistic are across samples
     • A systematic tendency for the statistic to over- or under-estimate the parameter
     • The chance that any single sample is collected improperly
     • The number of outliers present in one particular sample

     Correct answer: A systematic tendency for the statistic to over- or under-estimate the parameter

     Bias is a systematic tendency for the statistic to over- or under-estimate the parameter, meaning the center of the sampling distribution does not sit at the true parameter value. Spread is described by variability, not bias; how a single sample is collected is a separate sampling-method issue; and the count of outliers in one sample is unrelated to the long-run centering of the estimator.

158. Two estimators of the same parameter are both unbiased, but estimator A has a smaller standard deviation of its sampling distribution than estimator B. Which estimator is generally preferred and why?
     • Estimator B, because more variability gives more information
     • Estimator A, because lower variability means its values cluster more closely around the parameter
     • Neither, because unbiased estimators are always equally good
     • Estimator B, because higher variability reduces bias

     Correct answer: Estimator A, because lower variability means its values cluster more closely around the parameter

     Estimator A is preferred because lower variability means its values cluster more closely around the parameter, making individual estimates more reliable. More variability does not give more useful information about a fixed parameter, unbiased estimators are not automatically equal in quality, and variability does not affect bias since both estimators are already unbiased.

159. When sampling without replacement, which condition must be met so that the standard deviation formula for a statistic remains approximately valid?
     • The sample size must be at least 10% of the population
     • The sample size must be no more than 10% of the population
     • The population must be exactly twice the sample size
     • The population must be normally distributed

     Correct answer: The sample size must be no more than 10% of the population

     The sample size must be no more than 10% of the population, the so-called 10% condition, so that sampling without replacement does not appreciably change the standard deviation of the statistic. Requiring the sample to be at least 10% would violate this condition, a fixed population-to-sample ratio of two is not the rule, and normality of the population is a separate consideration from the 10% condition.

160. For the sampling distribution of a sample proportion to be approximately normal, which condition is checked using np and n(1 - p)?
     • The independence condition
     • The randomness condition
     • The large counts condition
     • The 10% condition

     Correct answer: The large counts condition

     The large counts condition is checked using np and n(1 - p), both of which should be at least 10 for the sampling distribution of the sample proportion to be approximately normal. The independence and 10% conditions concern whether observations affect one another, the randomness condition concerns how the sample was selected, and none of those use the np and n(1 - p) calculations.

161. A population proportion is p = 0.3 and a random sample of size n = 50 is taken. What is the standard deviation of the sampling distribution of the sample proportion?
     • Approximately 0.065
     • Approximately 0.30
     • Approximately 0.21
     • Approximately 0.0042

     Correct answer: Approximately 0.065

     The standard deviation of the sampling distribution of the sample proportion is approximately 0.065, found by taking $\sqrt{\frac{p(1 - p)}{n}}$, which is $\sqrt{\frac{(0.3)(0.7)}{50}}$. The value 0.30 is the proportion itself, 0.21 is p(1 - p) without dividing by n or taking the root, and 0.0042 is the variance before taking the square root.

162. Why must the population be normal (or the sample size large) for the sampling distribution of the sample mean to be normal?
     • Because only normal populations have a defined mean
     • Because small samples from non-normal populations may leave the sampling distribution non-normal
     • Because the sample mean is undefined for skewed data
     • Because randomness alone guarantees normality at any sample size

     Correct answer: Because small samples from non-normal populations may leave the sampling distribution non-normal

     Normality is required because small samples from non-normal populations may leave the sampling distribution non-normal; only a normal population or a large sample (via the central limit theorem) ensures the sampling distribution of the mean is approximately normal. Non-normal populations still have means, the sample mean is defined for skewed data, and randomness by itself does not produce normality at small sample sizes.

163. A population is exactly normal with mean 100 and standard deviation 15. For samples of size 9, what is the shape of the sampling distribution of the sample mean?
     • Skewed, because the sample size is small
     • Exactly normal, because the population is normal
     • Uniform, because all sample means are equally likely
     • Unknown, because the central limit theorem does not apply

     Correct answer: Exactly normal, because the population is normal

     The sampling distribution of the sample mean is exactly normal because the population is normal, and that holds for any sample size, including a small one like 9. A small sample does not introduce skew when the population itself is normal, sample means are not all equally likely, and the central limit theorem is not even needed since normality is inherited directly from a normal population.

164. What does the standard deviation of a sampling distribution measure?
     • The spread of individual data values within one sample
     • How much the statistic typically varies from sample to sample
     • The distance between the population mean and the population median
     • The number of samples needed to estimate the parameter

     Correct answer: How much the statistic typically varies from sample to sample

     The standard deviation of a sampling distribution measures how much the statistic typically varies from sample to sample, capturing the sampling variability of an estimate. It is not the spread of raw values within a single sample, not a gap between population center measures, and not a count of how many samples are required.

165. When constructing a sampling distribution for the difference between two sample proportions, what is the mean of that sampling distribution?
     • The sum of the two population proportions
     • The difference between the two population proportions
     • Always zero
     • The product of the two population proportions

     Correct answer: The difference between the two population proportions

     The mean of the sampling distribution of the difference between two sample proportions is the difference between the two population proportions, because each sample proportion is unbiased for its own population value. It is not their sum or product, and it is only zero in the special case where the two population proportions happen to be equal.

166. Two independent samples are taken to study the difference of two sample means. How is the standard deviation of the sampling distribution of the difference computed from the two individual sampling-distribution standard deviations?
     • By adding the two standard deviations directly
     • By taking $\sqrt{\sigma_1^2 + \sigma_2^2}$, the root of the sum of the two variances
     • By subtracting the smaller standard deviation from the larger
     • By averaging the two standard deviations

     Correct answer: By taking $\sqrt{\sigma_1^2 + \sigma_2^2}$, the root of the sum of the two variances

     The standard deviation of the sampling distribution of the difference is found by taking $\sqrt{\sigma_1^2 + \sigma_2^2}$, because variances of independent quantities add. Standard deviations themselves cannot be added or subtracted directly, and averaging them does not correctly combine independent variability.

167. A researcher uses a computer to repeatedly draw many random samples and record the sample mean of each. What is this process most directly used to approximate?
     • The population distribution of individual values
     • The sampling distribution of the sample mean
     • The exact value of the population mean
     • The bias of the sampling method

     Correct answer: The sampling distribution of the sample mean

     Repeatedly drawing many random samples and recording each sample mean is used to approximate the sampling distribution of the sample mean, since the collected means form a simulated version of that distribution. It does not reproduce the population distribution of individual values, does not pin down the exact population mean, and is not a measure of the sampling method's bias.

168. A simulated sampling distribution of sample means is centered very close to the true population mean but has visibly large spread. What does this suggest about the estimator?
     • It is biased but has low variability
     • It is approximately unbiased but has high variability
     • It is both biased and low in variability
     • It is neither unbiased nor variable

     Correct answer: It is approximately unbiased but has high variability

     Being centered near the true population mean indicates the estimator is approximately unbiased, while the large spread indicates high variability, so the estimator is approximately unbiased but has high variability. It is not biased since the center matches the parameter, and the visible spread rules out describing it as low in variability.

169. A population proportion is 0.6, and samples of size 100 are drawn. The large counts condition is checked with np = 60 and n(1 - p) = 40. What does this confirm?
     • The sampling distribution of the sample proportion can be treated as approximately normal
     • The population proportion must actually be 0.5
     • The samples were collected without randomness
     • The 10% condition is automatically satisfied

     Correct answer: The sampling distribution of the sample proportion can be treated as approximately normal

     Because np = 60 and n(1 - p) = 40 are both at least 10, the large counts condition is met, confirming the sampling distribution of the sample proportion can be treated as approximately normal. This calculation says nothing about forcing p to be 0.5, does not address how the samples were collected, and does not by itself verify the separate 10% condition.

170. Why does increasing the sample size reduce the standard deviation of a sampling distribution but not its bias?
     • Because bias depends on the centering of the estimator, which sample size does not change, while spread shrinks as n grows
     • Because larger samples always remove all bias automatically
     • Because sample size affects only the population mean, not the statistic
     • Because bias and standard deviation are the same quantity

     Correct answer: Because bias depends on the centering of the estimator, which sample size does not change, while spread shrinks as n grows

     Increasing sample size reduces spread because the standard deviation of a sampling distribution shrinks as n grows, but bias depends on the centering of the estimator, which sample size does not change. Larger samples do not automatically remove bias, sample size does not alter the fixed population mean, and bias and standard deviation are distinct concepts rather than the same quantity.

171. A one-proportion z-interval requires the standard error of the sample proportion. Which expression gives that standard error when the sample proportion is p-hat and the sample size is n?
     • N times p-hat times one minus p-hat
     • p-hat times one minus p-hat, divided by n, with no square root
     • $\sqrt{\frac{\hat{p}}{n}}$
     • $\sqrt{\frac{\hat{p}(1 - \hat{p})}{n}}$

     Correct answer: $\sqrt{\frac{\hat{p}(1 - \hat{p})}{n}}$

     The standard error for a confidence interval is $\sqrt{\frac{\hat{p}(1 - \hat{p})}{n}}$. The sample proportion is used because the true proportion is unknown when estimating. Omitting the square root leaves the variance rather than the standard error, dropping the failure term ignores variability from non-successes, and multiplying by n instead of dividing reverses how sample size affects spread.

172. When conducting a one-proportion significance test, the standard error in the test statistic is computed differently than in a confidence interval. Which value is used inside the standard error for the test?
     • The sample proportion p-hat
     • The hypothesized proportion p0 from the null hypothesis
     • The midpoint of p-hat and p0
     • The margin of error

     Correct answer: The hypothesized proportion p0 from the null hypothesis

     For a significance test the standard error uses the hypothesized proportion p0, because the test assumes the null hypothesis is true and builds the sampling distribution around p0. A confidence interval instead uses p-hat since it makes no null assumption. Averaging the two values or substituting the margin of error are not part of the correct test-statistic formula.

173. A one-proportion z test gives a test statistic of $z = 2.5$ for the alternative Ha: p > p0. Which statement best describes what this z-score represents?
     • The sample proportion equals 2.5
     • The probability of the result is 2.5
     • The observed proportion is 2.5 standard errors above the hypothesized proportion
     • The margin of error is 2.5

     Correct answer: The observed proportion is 2.5 standard errors above the hypothesized proportion

     A test statistic of $z = 2.5$ means the observed sample proportion lies 2.5 standard errors above the hypothesized proportion under the null model. The z-score standardizes the distance between p-hat and p0. It is not a proportion itself, not a probability such as a p-value, and not the margin of error, which belongs to interval estimation rather than the test statistic.

174. For the same data set, how does a 99% confidence interval for a proportion compare to a 90% confidence interval?
     • The 99% interval is centered at a different sample proportion
     • The 99% interval is narrower because higher confidence reduces uncertainty
     • The two intervals have identical widths
     • The 99% interval is wider because a higher confidence level uses a larger critical value

     Correct answer: The 99% interval is wider because a higher confidence level uses a larger critical value

     The 99% interval is wider because raising the confidence level increases the critical value z*, which enlarges the margin of error. Greater confidence requires capturing the parameter more often, so the interval must stretch farther. The interval does not become narrower with higher confidence, the widths differ, and both intervals share the same center at the sample proportion.

175. A researcher wants the margin of error for a proportion to be no more than 0.03 at 95% confidence and uses 0.5 for the planning proportion. Why is 0.5 chosen for this calculation?
     • It makes the math simplest to perform by hand
     • It produces the largest possible standard error, giving a conservative sample size
     • It is always the true population proportion
     • It minimizes the required sample size

     Correct answer: It produces the largest possible standard error, giving a conservative sample size

     Using 0.5 is chosen because the product p times one minus p is maximized at p equal to 0.5, which yields the largest standard error and therefore the most conservative, or largest, required sample size. This guards against underestimating the needed sample when the true proportion is unknown. It is not selected for arithmetic ease, is not assumed to be the true value, and it maximizes rather than minimizes the sample size.

176. A 95% confidence interval for the proportion of voters favoring a tax increase runs from 0.42 to 0.49. Based on this interval, what can be concluded about the claim that a majority favor the increase?
     • There is convincing evidence that a majority favor the increase
     • The interval gives no information about the majority claim
     • There is convincing evidence that a majority do not favor the increase, since the interval lies entirely below 0.50
     • A majority is plausible because 0.50 is close to the interval

     Correct answer: There is convincing evidence that a majority do not favor the increase, since the interval lies entirely below 0.50

     Because the entire interval from 0.42 to 0.49 falls below 0.50, there is convincing evidence that a majority do not favor the increase. A confidence interval can be used to assess a claim about a value: since 0.50 is not a plausible value, the majority claim is not supported. The interval clearly addresses the claim, and 0.50 being nearby does not make it plausible when it sits outside the interval.

177. In a two-proportion significance test of H0: p1 = p2, the data from both groups are combined to estimate a single proportion. What is this quantity called and why is it used?
     • The combined proportion, used to widen the confidence interval
     • The pooled proportion, used because the null assumes the two population proportions are equal
     • The average proportion, used to compute the margin of error
     • The critical proportion, used to set the significance level

     Correct answer: The pooled proportion, used because the null assumes the two population proportions are equal

     It is called the pooled proportion, and it is used because the null hypothesis assumes the two population proportions are equal, so all successes and failures are combined to estimate that common value. This pooled estimate goes into the standard error of the test. It is not used to widen an interval or set alpha, and it is more than a simple average since it weights by group sizes.

178. Which of the following is the correct point estimate reported in a two-proportion confidence interval comparing groups 1 and 2?
     • The pooled proportion combining both groups
     • The ratio of the two sample proportions
     • The sum of the two sample proportions
     • The difference between the two sample proportions, p-hat 1 minus p-hat 2

     Correct answer: The difference between the two sample proportions, p-hat 1 minus p-hat 2

     The point estimate for a two-proportion confidence interval is the difference between the two sample proportions, p-hat 1 minus p-hat 2, because the interval estimates the difference in population proportions. The pooled proportion is used in a significance test, not in the interval estimate. A sum or ratio of proportions is not the parameter being estimated when comparing two groups by a difference.

179. A two-proportion z-interval for p1 minus p2 is computed as -0.02 to 0.11. What does the inclusion of 0 in this interval indicate?
     • The two population proportions are definitely equal
     • Group 1 has a higher proportion than group 2
     • There is no plausible difference, so it is plausible the two proportions are equal
     • The interval was computed incorrectly

     Correct answer: There is no plausible difference, so it is plausible the two proportions are equal

     Because 0 lies within the interval from -0.02 to 0.11, it is plausible that the two population proportions are equal, since a difference of zero is among the plausible values. This does not prove the proportions are exactly equal, only that no difference is plausible. It does not establish that group 1 is higher, and the presence of 0 is a normal, correct outcome, not an error.

180. Before running a two-proportion z-test, which condition must be checked regarding the counts in each group?
     • Each group must have at least 10 successes and at least 10 failures
     • Only the total combined sample size must exceed 30
     • Only one of the two groups needs at least 10 successes
     • The two sample proportions must be exactly equal

     Correct answer: Each group must have at least 10 successes and at least 10 failures

     The large-counts condition requires that each group separately have at least 10 successes and at least 10 failures, so the sampling distribution of each proportion is approximately normal. Checking only the combined total or just one group fails to validate the normal approximation for both. The sample proportions need not be equal; in fact the test exists precisely to compare them.

181. A significance test for a proportion uses data from a sample collected without randomization. How does this affect the validity of the conclusions?
     • It has no effect as long as the sample is large
     • It undermines the ability to generalize results, since the randomness condition is not met
     • It guarantees a Type I error will occur
     • It automatically makes the p-value smaller

     Correct answer: It undermines the ability to generalize results, since the randomness condition is not met

     A lack of randomization undermines generalizing the conclusions because the randomness condition, required for valid inference, is not satisfied. Inference procedures assume data come from a random sample or randomized experiment. Large sample size does not repair selection bias, the missing randomness does not guarantee a specific error, and it does not predictably change the p-value's size.

182. A test yields a p-value of 0.08. Using a significance level of 0.10, what decision is made, and using 0.05, what decision is made?
     • Reject the null at 0.10 but fail to reject at 0.05
     • Fail to reject at both 0.10 and 0.05
     • Reject the null at both 0.10 and 0.05
     • Reject the null at 0.05 but fail to reject at 0.10

     Correct answer: Reject the null at 0.10 but fail to reject at 0.05

     With a p-value of 0.08, the null is rejected at the 0.10 level because 0.08 is less than 0.10, but it is not rejected at the 0.05 level because 0.08 is greater than 0.05. The decision depends on comparing the p-value to the chosen significance level. The same p-value can lead to different conclusions at different alpha levels, which is why both comparisons matter.

183. How is the significance level alpha related to a Type I error in a significance test?
     • Alpha equals the probability of a Type II error
     • Alpha equals the probability of a Type I error when the null is true
     • Alpha equals the power of the test
     • Alpha equals the p-value of the test

     Correct answer: Alpha equals the probability of a Type I error when the null is true

     The significance level alpha is the probability of making a Type I error, that is, rejecting a true null hypothesis. By choosing alpha, the researcher sets the acceptable false-positive rate before collecting data. Alpha is not the probability of a Type II error, not the power, and not the p-value, which is computed from the observed data rather than chosen in advance.

184. All else held constant, which change would increase the power of a one-proportion significance test?
     • Decreasing the sample size
     • Lowering the significance level alpha
     • Increasing the sample size
     • Making the true proportion closer to the hypothesized value

     Correct answer: Increasing the sample size

     Increasing the sample size raises power because a larger sample reduces variability and makes it easier to detect a true difference from the hypothesized value. Decreasing sample size lowers power, lowering alpha makes rejection harder and reduces power, and a true proportion closer to the null value makes the effect harder to detect. Only a larger sample reliably increases power here.

185. A medical team concludes a treatment is effective when in reality it has no effect. In terms of patient consequences, which type of error has occurred?
     • A Type II error, leading to a missed effective treatment
     • A Type I error, leading to use of an ineffective treatment
     • No error, because the conclusion favored action
     • A sampling error unrelated to hypothesis testing

     Correct answer: A Type I error, leading to use of an ineffective treatment

     Concluding the treatment works when it truly does not is a Type I error, since a true null of no effect was wrongly rejected. The practical consequence is using a treatment that provides no benefit. A Type II error would be failing to detect a real effect, which is the opposite situation, and the mistake is a genuine inference error, not just sampling noise.

186. A confidence interval and a two-sided significance test at the same confidence level can lead to consistent conclusions. If a 95% confidence interval for p does not contain the value p0, what would a two-sided test of H0: p = p0 at alpha = 0.05 conclude?
     • Fail to reject the null hypothesis
     • The two methods cannot be compared
     • Reject the null hypothesis
     • Accept the null hypothesis as true

     Correct answer: Reject the null hypothesis

     If a 95% confidence interval excludes p0, then a two-sided test at the 0.05 level rejects the null hypothesis, because p0 is not a plausible value for the parameter. The duality between intervals and two-sided tests means values outside the interval would be rejected. The methods are directly comparable, and one never accepts the null as proven true.

187. A school claims at least 80% of graduates are employed within a year. To test this skeptically, which alternative hypothesis matches an investigator doubting the claim?
     • Ha: p > 0.80
     • Ha: p = 0.80
     • Ha: p < 0.80
     • Ha: p-hat < 0.80

     Correct answer: Ha: p < 0.80

     An investigator doubting the claim of at least 80% employment would use Ha: p < 0.80, testing whether the true proportion falls short of the claimed level. The alternative reflects the suspicion that employment is lower than stated. A greater-than alternative tests the opposite direction, the alternative never contains an equality, and hypotheses are written about the parameter p rather than the statistic p-hat.

188. Two news outlets report the same poll. Outlet A states a margin of error of plus or minus 2 percentage points and outlet B states plus or minus 4 percentage points for the same proportion. Assuming equal sample sizes, what most likely explains the difference?
     • Outlet B used a higher confidence level than outlet A
     • Outlet A used a higher confidence level than outlet B
     • Outlet A used a larger sample
     • Outlet B reported the point estimate incorrectly

     Correct answer: Outlet B used a higher confidence level than outlet A

     With equal sample sizes, outlet B's larger margin of error most likely results from using a higher confidence level, which increases the critical value and widens the margin. A higher confidence level for outlet A would shrink, not widen, its margin relative to B. Equal sample sizes rule out a sample-size explanation, and a misreported point estimate would not change the margin of error.

189. Why must inference for a proportion be based on a categorical variable rather than a quantitative one?
     • Quantitative variables cannot be sampled randomly
     • Proportions are computed by averaging numerical measurements
     • Categorical variables always produce larger samples
     • Proportions summarize the share of observations falling into a category

     Correct answer: Proportions summarize the share of observations falling into a category

     Inference for a proportion uses a categorical variable because a proportion represents the share of observations falling into a particular category, such as success versus failure. Quantitative data are summarized by means, not proportions, so averaging numerical measurements describes a mean instead. Sample size and randomization apply to both variable types, so those reasons do not distinguish proportions.

190. A study finds a statistically significant difference between two proportions but the confidence interval for the difference is very close to 0, such as 0.001 to 0.012. What does this illustrate?
     • Statistical significance always means a large practical effect
     • The proportions are exactly equal
     • The test must have been performed incorrectly
     • A result can be statistically significant yet small in practical importance

     Correct answer: A result can be statistically significant yet small in practical importance

     This illustrates that a result can be statistically significant while the actual difference is small in practical importance, since a narrow interval near 0 shows the effect, though real, is tiny. Large samples can detect minuscule differences. Significance does not guarantee a large effect, the test is not necessarily flawed, and an interval excluding 0 means the proportions are not exactly equal.

191. When the conditions for a one-proportion z-test are met, which distribution is used to find the p-value?
     • The standard normal distribution
     • The t-distribution with n minus 1 degrees of freedom
     • The $\chi^2$ distribution
     • The binomial distribution with parameter p-hat

     Correct answer: The standard normal distribution

     The p-value for a one-proportion z-test is found using the standard normal distribution, because the standardized test statistic follows an approximately normal model when the large-counts condition holds. The t-distribution applies to inference about means with unknown standard deviation, not proportions. The $\chi^2$ distribution is used for categorical tables, and the exact binomial is not the basis of the z-procedure.

192. A two-sided test of H0: p = 0.4 produces a sample proportion below 0.4. How is the two-sided p-value related to the one-sided tail area in this case?
     • It is double the one-sided tail area
     • It equals the one-sided tail area
     • It is half the one-sided tail area
     • It is unrelated to the tail area

     Correct answer: It is double the one-sided tail area

     For a two-sided test, the p-value is double the one-sided tail area because both tails of the distribution count as results at least as extreme as the observed one. Since the alternative allows deviations in either direction, the area in the matching tail is doubled. It is not equal to or half of a single tail, and it is directly built from the tail area, not unrelated.

193. A researcher increases the sample size in a one-proportion confidence interval while keeping the confidence level and sample proportion the same. What happens to the width of the interval?
     • It increases
     • It becomes zero
     • It stays exactly the same
     • It decreases

     Correct answer: It decreases

     Increasing the sample size decreases the width of the interval because a larger n shrinks the standard error, reducing the margin of error on each side. With the confidence level and sample proportion unchanged, only the standard error changes. The width does not increase or remain fixed with a larger sample, and it approaches but never reaches zero for any finite sample.

194. In reporting the conclusion of a one-proportion significance test, why is it incorrect to say the test proves the null hypothesis is true when failing to reject it?
     • Failing to reject means there is not enough evidence against the null, not proof it is true
     • Failing to reject always means a Type I error occurred
     • The null hypothesis can never be written about a proportion
     • Proving the null requires a larger significance level

     Correct answer: Failing to reject means there is not enough evidence against the null, not proof it is true

     Failing to reject the null means only that there is insufficient evidence against it, not that it is proven true, because the test can never confirm a parameter exactly. A non-significant result is consistent with the null but also with small effects the test could not detect. It does not imply a Type I error, the null is routinely about a proportion, and no alpha level can prove the null.

195. In a one-sample t interval for a population mean, what does the margin of error represent?
     • The difference between the largest and smallest values in the sample
     • The product of the critical t value and the standard error of the mean
     • The hypothesized value of the population mean
     • The probability that the interval misses the true mean

     Correct answer: The product of the critical t value and the standard error of the mean

     The margin of error in a one-sample t interval is the product of the critical t value and the standard error of the mean, and it is added to and subtracted from the sample mean to form the interval. It is not the range of the data, not the hypothesized mean, and not a probability, since those quantities describe other features of the data or procedure.

196. A 95% confidence interval for a population mean is reported as 48 plus or minus 6. What is the point estimate of the population mean?
     • 6
     • 48
     • 42
     • 54

     Correct answer: 48

     The point estimate is 48, because in an interval written as a center plus or minus a margin of error, the center value is the sample mean that serves as the point estimate. The value 6 is the margin of error, while 42 and 54 are the lower and upper endpoints of the interval rather than the point estimate.

197. A teacher wants to estimate the mean score on a final exam with a smaller margin of error while keeping the same confidence level. Which action will accomplish this?
     • Increase the sample size
     • Decrease the sample size
     • Raise the confidence level
     • Use a one-sided interval interpretation

     Correct answer: Increase the sample size

     Increasing the sample size will shrink the margin of error while holding the confidence level constant, because a larger sample lowers the standard error of the mean. Decreasing the sample size would enlarge the margin of error, raising the confidence level would also widen it, and switching to a one-sided interpretation does not address the goal of a smaller two-sided margin.

198. For a one-sample t procedure with a sample of size 18, how many degrees of freedom are used to find the critical value?
     • 17
     • 18
     • 19
     • 9

     Correct answer: 17

     The degrees of freedom equal 17, found by subtracting one from the sample size of 18, since a one-sample t procedure uses n minus 1 degrees of freedom. The value 18 ignores the subtraction, 19 adds instead of subtracts, and 9 incorrectly halves the sample size.

199. When the conservative approach is used to find degrees of freedom for a two-sample t procedure by hand, which value is typically chosen?
     • The sum of the two sample sizes
     • The smaller of the two sample sizes minus one
     • The larger of the two sample sizes minus one
     • The product of the two sample sizes

     Correct answer: The smaller of the two sample sizes minus one

     The conservative approach uses the smaller of the two sample sizes minus one for the degrees of freedom, which produces a larger critical value and a wider, safer interval. Summing the sample sizes overstates the degrees of freedom, using the larger sample size is not conservative, and multiplying the sample sizes does not correspond to any degrees of freedom rule.

200. A matched-pairs t test is most appropriate when the data consist of which of the following?
     • Two independent random samples from two separate populations
     • Counts of successes and failures in two groups
     • A single categorical variable measured once
     • Pairs of related observations, with the analysis performed on the differences within each pair

     Correct answer: Pairs of related observations, with the analysis performed on the differences within each pair

     A matched-pairs t test is appropriate when the data consist of pairs of related observations, and the test is carried out on the differences within each pair. Two independent samples call for a two-sample procedure instead, a single categorical variable is not quantitative, and counts of successes and failures call for proportion-based methods.

201. Which set of hypotheses is appropriate for a paired t test where the mean difference is denoted mu sub d?
     • H0: p = 0 versus Ha: p is not equal to 0
     • H0: sigma = 0 versus Ha: sigma is not equal to 0
     • H0: x-bar = 0 versus Ha: x-bar is not equal to 0
     • H0: mu sub d = 0 versus Ha: mu sub d is not equal to 0

     Correct answer: H0: mu sub d = 0 versus Ha: mu sub d is not equal to 0

     The appropriate hypotheses are H0: mu sub d = 0 versus Ha: mu sub d is not equal to 0, because a paired t test concerns the mean of the population of differences. Using p refers to a proportion, using x-bar incorrectly states the hypothesis about a statistic rather than a parameter, and using sigma refers to a standard deviation rather than a mean difference.

202. In a two-sample t test for the difference of two means, what is the null hypothesis usually stated as?
     • One population mean is twice the other
     • The two population means differ by exactly 10
     • The two sample means are equal
     • The two population means are equal

     Correct answer: The two population means are equal

     The null hypothesis for a two-sample t test is usually that the two population means are equal, equivalently that their difference is zero. The hypothesis is about population parameters, not the observed sample means, and it does not by default specify a difference of 10 or a doubling relationship unless a problem explicitly states such a claim.

203. A researcher calculates a one-sample t statistic of 2.40 for a test of H0: mu = 50 against Ha: mu greater than 50. What does the positive sign of the test statistic indicate?
     • The sample mean is below the hypothesized value of 50
     • The sample size is large
     • The population standard deviation is positive
     • The sample mean is above the hypothesized value of 50

     Correct answer: The sample mean is above the hypothesized value of 50

     A positive t statistic indicates the sample mean is above the hypothesized value of 50, because the numerator is the sample mean minus the hypothesized mean. A sample mean below 50 would yield a negative statistic, while the sign of the statistic is unrelated to whether the standard deviation is positive or to the size of the sample.

204. Which of the following is the correct interpretation of a p-value in a one-sample t test for a mean?
     • The probability that the null hypothesis is true
     • The probability that the alternative hypothesis is true
     • The probability of getting a sample result at least as extreme as the one observed, assuming the null hypothesis is true
     • The proportion of the population that lies within the confidence interval

     Correct answer: The probability of getting a sample result at least as extreme as the one observed, assuming the null hypothesis is true

     The p-value is the probability of obtaining a sample result at least as extreme as the one observed, assuming the null hypothesis is true. It is not the probability that either hypothesis is true, since hypotheses are not assigned probabilities in this framework, and it is not a statement about the share of the population inside an interval.

205. A 99% confidence interval for the mean weight of a product is (245, 255) grams. A quality inspector claims the mean is 250 grams. Based on this interval, what should the inspector conclude?
     • The claim is plausible because 250 is inside the interval
     • The claim is contradicted because 250 is outside the interval
     • The interval proves the mean is exactly 250 grams
     • The confidence level must be lowered to test the claim

     Correct answer: The claim is plausible because 250 is inside the interval

     The claim is plausible because 250 grams lies inside the interval from 245 to 255, and any value within a confidence interval is consistent with the data at that confidence level. The value 250 is not outside the interval, the interval cannot prove an exact value, and there is no need to change the confidence level to assess this claim.

206. A study reports a 95% confidence interval for the difference in mean reaction times (treatment minus control) as (-0.8, 0.5) seconds. What does this interval suggest about the difference in means?
     • There is no convincing evidence of a difference, since the interval includes zero
     • There is convincing evidence the treatment mean is lower
     • There is convincing evidence the treatment mean is higher
     • The two means are exactly equal

     Correct answer: There is no convincing evidence of a difference, since the interval includes zero

     Because the interval from -0.8 to 0.5 includes zero, there is no convincing evidence of a difference between the two means at the 95% confidence level. An interval that contains zero allows for the possibility of no difference, so it neither establishes that one mean is higher or lower nor proves the means are exactly equal.

207. In a one-sample t test, a Type I error occurs when which of the following happens?
     • The null hypothesis is rejected when it is actually true
     • The null hypothesis is not rejected when it is actually false
     • The sample mean equals the population mean
     • The confidence level is set too high

     Correct answer: The null hypothesis is rejected when it is actually true

     A Type I error occurs when the null hypothesis is rejected even though it is actually true, producing a false positive conclusion about the mean. Failing to reject a false null is a Type II error, the sample mean matching the population mean is not an error, and the choice of confidence level is a design decision rather than an error.

208. For a fixed effect size in a test about a population mean, increasing the significance level alpha generally has what effect on the power of the test?
     • It decreases the power
     • It has no effect on the power
     • It increases the power
     • It makes the power exactly equal to alpha

     Correct answer: It increases the power

     Increasing the significance level alpha generally increases the power of a test about a mean, because a larger rejection region makes it easier to reject a false null hypothesis. Raising alpha does not decrease power, it does have an effect, and power is not set equal to alpha, since power measures the chance of correctly rejecting a false null.

209. A nutrition researcher computes a one-sample t interval but the sample of size 12 shows a strong skew with a clear outlier on a graph. What is the main concern for the inference?
     • The interval will be too narrow to be useful
     • The degrees of freedom will be negative
     • The normality condition may be violated, making the t interval unreliable for this small sample
     • The sample mean cannot be computed

     Correct answer: The normality condition may be violated, making the t interval unreliable for this small sample

     The main concern is that the normality condition may be violated, making the t interval unreliable, because a small sample of size 12 with strong skew and an outlier suggests the population may not be normal. The skew does not force the interval to be too narrow, degrees of freedom cannot be negative, and the sample mean can always be computed from the data.

210. A two-sample t interval for the difference in mean heights of two plant varieties (A minus B) is (1.2, 3.6) centimeters. What does this interval indicate?
     • Variety A and variety B have equal mean heights
     • Variety A has a higher mean height than variety B
     • Variety B has a higher mean height than variety A
     • No comparison of the means can be made

     Correct answer: Variety A has a higher mean height than variety B

     The interval from 1.2 to 3.6 centimeters indicates variety A has a higher mean height than variety B, because the entire interval for A minus B lies above zero. Since the interval does not include zero, the means are not plausibly equal, and a positive interval rules out variety B being taller and clearly allows a comparison.

211. An experimenter measures the cholesterol of 30 patients before and after a medication and runs a paired t test. How many degrees of freedom does this test use?
     • 60
     • 58
     • 29
     • 30

     Correct answer: 29

     The paired t test uses 29 degrees of freedom, found by subtracting one from the number of pairs, which is 30 pairs minus one. A paired analysis reduces to a one-sample test on the 30 differences, so the count is not based on 60 total measurements, and the values 60 and 58 incorrectly treat the data as two independent groups.

212. Why is a paired design often preferred over two independent samples when the same subjects can be measured under two conditions?
     • It eliminates the need to check any conditions for inference
     • It always produces a larger p-value
     • It reduces variability by controlling for differences between subjects
     • It removes the need to compute a sample mean

     Correct answer: It reduces variability by controlling for differences between subjects

     A paired design is often preferred because it reduces variability by controlling for differences between subjects, since each subject serves as their own comparison. It does not remove the need to check inference conditions, it does not guarantee a larger p-value, and computing a mean of the differences is still required.

213. A one-sample t test gives a p-value of 0.12 at a significance level of 0.05. What is the correct conclusion?
     • Reject the null hypothesis; there is convincing evidence about the mean
     • The test should be repeated until the p-value is below 0.05
     • Accept the alternative hypothesis as proven
     • Fail to reject the null hypothesis; there is not convincing evidence about the mean

     Correct answer: Fail to reject the null hypothesis; there is not convincing evidence about the mean

     Because the p-value of 0.12 exceeds the significance level of 0.05, we fail to reject the null hypothesis and conclude there is not convincing evidence about the mean. A p-value above alpha does not justify rejecting the null, we never accept the alternative as proven, and repeating a test just to chase significance is not a valid procedure.

214. What is the standard error of the difference of two sample means built from in a two-sample t procedure?
     • The sum of the two sample standard deviations
     • $\sqrt{\frac{s_1^2}{n_1} + \frac{s_2^2}{n_2}}$
     • The difference of the two sample means
     • The larger sample standard deviation divided by the total sample size

     Correct answer: $\sqrt{\frac{s_1^2}{n_1} + \frac{s_2^2}{n_2}}$

     The standard error of the difference of two sample means is $\sqrt{\frac{s_1^2}{n_1} + \frac{s_2^2}{n_2}}$, because the variances of independent samples add. Adding the standard deviations directly is incorrect, the difference of the means is the estimate rather than its standard error, and using only the larger standard deviation ignores the second group.

215. A sample of 16 measurements has a mean of 25 and a sample standard deviation of 4. What is the standard error of the mean for a one-sample t procedure?
     • 4
     • 0.25
     • 1
     • 16

     Correct answer: 1

     The standard error of the mean is 1, found by dividing the sample standard deviation of 4 by $\sqrt{16}$, which is 4 divided by 4. The value 4 is the standard deviation itself, 0.25 divides incorrectly by the sample size, and 16 is the sample size rather than a standard error.

216. In hypothesis testing for a mean, what does the significance level alpha represent?
     • The probability of a Type II error
     • The probability of correctly failing to reject a true null
     • The power of the test
     • The maximum probability of a Type I error that the researcher is willing to accept

     Correct answer: The maximum probability of a Type I error that the researcher is willing to accept

     The significance level alpha represents the maximum probability of a Type I error that the researcher is willing to accept, setting the threshold for rejecting a true null hypothesis. It is not the probability of a Type II error, not the chance of correctly retaining a true null, and not the power of the test, which measures detecting a false null.

217. A study comparing the mean income of two cities uses independent random samples. Which condition specifically addresses whether one sample's values could influence the other's?
     • The randomness condition
     • The symmetry condition
     • The large counts condition
     • The independence condition, including independence between the two groups

     Correct answer: The independence condition, including independence between the two groups

     The independence condition, including independence between the two groups, addresses whether the values in one sample could influence the values in the other, and it must hold for a valid two-sample t procedure. The randomness condition concerns how samples are selected, the large counts condition applies to proportions, and there is no separate symmetry condition for this procedure.

218. A 90% confidence interval for a population mean is (62, 78). What is the margin of error for this interval?
     • 8
     • 16
     • 70
     • 62

     Correct answer: 8

     The margin of error is 8, found by taking half the width of the interval, since the width from 62 to 78 is 16 and the margin is half of that. The full width 16 is twice the margin, 70 is the center of the interval, and 62 is the lower endpoint rather than the margin.

219. When the population standard deviation is unknown and a sample standard deviation is used instead, why is the t distribution preferred over the normal distribution for inference about a mean?
     • Because the t distribution has a smaller spread than the normal distribution
     • Because the t distribution accounts for the extra uncertainty from estimating the standard deviation
     • Because the t distribution is only defined for large samples
     • Because the t distribution requires the population to be uniform

     Correct answer: Because the t distribution accounts for the extra uncertainty from estimating the standard deviation

     The t distribution is preferred because it accounts for the extra uncertainty introduced by estimating the standard deviation from the sample, giving it heavier tails than the normal distribution. The t distribution has a larger spread, not a smaller one, it is defined for all sample sizes through its degrees of freedom, and it does not require a uniform population.

220. As the degrees of freedom for a t distribution increase, what happens to its shape?
     • It becomes more skewed to the right
     • It becomes flatter with heavier tails
     • It approaches the standard normal distribution
     • It becomes a uniform distribution

     Correct answer: It approaches the standard normal distribution

     As the degrees of freedom increase, the t distribution approaches the standard normal distribution, because the extra uncertainty from estimating the standard deviation shrinks with larger samples. The t distribution is symmetric rather than skewed, its tails grow lighter rather than heavier as degrees of freedom rise, and it never becomes uniform.

221. A car company tests whether a new tire design changes braking distance compared with the old design using the same 25 cars under both designs. Which procedure is appropriate?
     • A two-sample t test treating the two designs as independent groups
     • A matched-pairs t test on the differences in braking distance
     • A test for a single population proportion
     • A two-proportion z test

     Correct answer: A matched-pairs t test on the differences in braking distance

     A matched-pairs t test on the differences in braking distance is appropriate because the same 25 cars are measured under both tire designs, creating paired data. Treating the designs as independent groups ignores the natural pairing, and proportion tests do not apply to the quantitative braking-distance measurements.

222. A one-sample t interval for a mean assumes the data come from a random sample. What goes wrong with the inference if the sample was actually a voluntary response sample?
     • Bias may make the interval fail to capture the true population mean reliably
     • The standard error becomes negative
     • The degrees of freedom must be recalculated
     • The confidence level automatically rises to 100%

     Correct answer: Bias may make the interval fail to capture the true population mean reliably

     If the data come from a voluntary response sample, bias may make the interval fail to capture the true population mean reliably, because the sample may not represent the population. Degrees of freedom depend only on sample size, standard errors are never negative, and a flawed sampling method does not raise the confidence level to 100%.

223. A two-sample t test produces a test statistic of $t = -3.1$ for comparing mean test scores (group 1 minus group 2). What does the negative value indicate?
     • The mean of group 1 is less than the mean of group 2
     • The mean of group 1 is greater than the mean of group 2
     • The two sample means are equal
     • The standard error is negative

     Correct answer: The mean of group 1 is less than the mean of group 2

     A negative test statistic indicates the mean of group 1 is less than the mean of group 2, because the numerator computes the group 1 mean minus the group 2 mean. A positive difference would give a positive statistic, equal means would give a statistic near zero, and a standard error is always positive.

224. A 95% confidence interval and a two-sided significance test at alpha = 0.05 are used on the same data for a mean. If the hypothesized value lies outside the 95% interval, what is the corresponding test conclusion?
     • Fail to reject the null hypothesis
     • Reject the null hypothesis
     • The two methods give contradictory results
     • No conclusion is possible without the p-value

     Correct answer: Reject the null hypothesis

     If the hypothesized value lies outside the 95% confidence interval, the corresponding two-sided test at alpha = 0.05 will reject the null hypothesis, because the interval and the test are linked. A value outside the interval is not consistent with the data, so we do not fail to reject, the methods agree rather than contradict, and the interval alone supports the conclusion.

225. A researcher wants to detect a smaller true difference between two population means without changing alpha. Which change increases the power of the two-sample t test?
     • Decreasing both sample sizes
     • Increasing the variability within each group
     • Increasing both sample sizes
     • Reducing the true difference even further

     Correct answer: Increasing both sample sizes

     Increasing both sample sizes increases the power of the two-sample t test, because larger samples reduce the standard error and make a true difference easier to detect. Decreasing sample sizes lowers power, more within-group variability lowers power, and a smaller true difference is harder, not easier, to detect.

226. A confidence interval for a mean is computed at the 80% confidence level. Compared with a 95% interval from the same data, the 80% interval will be which of the following?
     • Wider, because lower confidence requires a larger critical value
     • Centered at a higher value
     • Identical, because the data are the same
     • Narrower, because lower confidence requires a smaller critical value

     Correct answer: Narrower, because lower confidence requires a smaller critical value

     The 80% interval will be narrower because a lower confidence level uses a smaller critical t value, producing a smaller margin of error. Lower confidence does not require a larger critical value, the intervals differ despite using the same data, and both intervals share the same center at the sample mean.

227. A one-sample t test is conducted with Ha: mu less than 30 (a left-tailed test). For which kind of sample mean will the p-value be small?
     • A sample mean much greater than 30
     • A sample mean exactly equal to 30
     • A sample mean much less than 30
     • Any sample mean, regardless of its value

     Correct answer: A sample mean much less than 30

     For a left-tailed test with Ha: mu less than 30, the p-value will be small when the sample mean is much less than 30, since that result provides strong evidence in the direction of the alternative. A sample mean above 30 points away from the alternative, a mean equal to 30 gives a large p-value, and the p-value clearly depends on the sample mean.

228. What is the correct way to interpret a 95% confidence level for a method that produces intervals for a mean?
     • In repeated sampling, about 95% of the intervals constructed this way will capture the true mean
     • Each individual interval has a 95% chance of being correct after it is computed
     • 95% of the population values fall inside any one interval
     • The sample mean equals the population mean 95% of the time

     Correct answer: In repeated sampling, about 95% of the intervals constructed this way will capture the true mean

     The 95% confidence level means that in repeated sampling, about 95% of the intervals constructed by this method will capture the true mean. It is not a probability statement about a single completed interval, it does not describe the share of population values inside one interval, and it does not claim the sample mean equals the population mean most of the time.

229. A two-sample t test requires that the two samples be independent. Which of the following study designs satisfies this requirement?
     • Measuring the same people twice under two conditions
     • Measuring one group at two different time points
     • Pairing each subject with a similar partner before measuring
     • Randomly assigning different subjects to two separate treatment groups

     Correct answer: Randomly assigning different subjects to two separate treatment groups

     Randomly assigning different subjects to two separate treatment groups satisfies the independence requirement for a two-sample t test, because the two groups consist of distinct, unrelated individuals. Measuring the same people twice, pairing subjects, or measuring one group at two times all create dependent data better suited to a paired procedure.

230. A confidence interval for the difference of two means is found to be entirely negative, such as (-5.0, -1.0). What does this indicate about the two population means (computed as mean 1 minus mean 2)?
     • Mean 1 is plausibly larger than mean 2
     • Mean 1 is plausibly smaller than mean 2
     • The two means are plausibly equal
     • The difference is plausibly zero

     Correct answer: Mean 1 is plausibly smaller than mean 2

     An interval that is entirely negative, such as -5.0 to -1.0, indicates mean 1 is plausibly smaller than mean 2, because the difference mean 1 minus mean 2 is below zero throughout. A positive difference would mean the first mean is larger, and since zero is not in the interval the means are not plausibly equal nor is the difference plausibly zero.

231. In a paired t procedure, what quantity is actually analyzed to test for a mean difference?
     • The two original groups of measurements treated separately
     • The set of within-pair differences
     • The ratio of the two measurements in each pair
     • The combined list of all measurements pooled together

     Correct answer: The set of within-pair differences

     A paired t procedure analyzes the set of within-pair differences, reducing the problem to a one-sample t analysis on those differences. It does not treat the two groups separately, it does not use the ratio of paired measurements, and it does not simply pool all measurements together, since pooling would discard the pairing structure.

232. A researcher reduces the significance level for a test about a mean from 0.05 to 0.01 to be more cautious. Holding everything else fixed, what is one consequence of this change?
     • The probability of a Type I error decreases but the probability of a Type II error increases
     • The probability of a Type I error increases
     • Both error probabilities decrease at once
     • The power of the test increases

     Correct answer: The probability of a Type I error decreases but the probability of a Type II error increases

     Lowering the significance level from 0.05 to 0.01 decreases the probability of a Type I error but increases the probability of a Type II error, because a stricter threshold makes the null harder to reject. The Type I error probability does not increase, both error probabilities cannot drop at once from this single change, and the power decreases rather than increases.