AP Statistics Flashcards: Complete 5-Unit Course Review

Cards in this deck

1. Card 1

   Question

   What makes a question a statistical investigative question?

   Answer

   It anticipates variability in data and can be answered by collecting and analyzing data about a population or process.

2. Card 2

   Question

   What is an observational unit?

   Answer

   An individual item or person from which data are collected.

3. Card 3

   Question

   A student's class year is recorded as freshman, sophomore, junior, or senior. What type of variable is this?

   Answer

   Categorical. The values name groups rather than measure a numerical amount.

4. Card 4

   Question

   How does a parameter differ from a statistic?

   Answer

   A parameter describes a population; a statistic describes a sample.

5. Card 5

   Question

   How is a category's relative frequency calculated?

   Answer

   Divide the category count by the total number of observations.

6. Card 6

   Question

   What should the height of a bar represent in a relative-frequency bar chart?

   Answer

   The proportion or percentage of observations in that category.

7. Card 7

   Question

   Number of text messages sent in a day: discrete or continuous?

   Answer

   Discrete. It is a count with separated possible values.

8. Card 8

   Question

   Which displays preserve individual quantitative data values?

   Answer

   Dotplots and stem-and-leaf plots. A histogram groups values into intervals.

9. Card 9

   Question

   What four features should a description of a quantitative distribution address?

   Answer

   Shape, center, variability, and unusual features such as gaps or outliers.

10. Card 10

    Question

    Which measure of center is usually better for a strongly right-skewed distribution?

    Answer

    The median, because it is resistant to extreme high values.

11. Card 11

    Question

    The values are 3, 5, 5, and 11. What is the mean?

    Answer

    6. The sum is 24, divided by 4 observations.

12. Card 12

    Question

    The ordered values are 2, 4, 7, 9, 12, and 20. What is the median?

    Answer

    8, the average of the two middle values 7 and 9.

13. Card 13

    Question

    How is the interquartile range calculated?

    Answer

    IQR = Q3 − Q1. It measures the spread of the middle 50% of the data.

14. Card 14

    Question

    What does a small standard deviation say about a data set?

    Answer

    Values typically lie close to the mean.

15. Card 15

    Question

    Which common summaries are resistant to extreme values?

    Answer

    The median and IQR are resistant; the mean and standard deviation are not.

16. Card 16

    Question

    In a modified boxplot, where do the whiskers end?

    Answer

    At the smallest and largest observed values within the 1.5 × IQR fences; values beyond the fences are plotted separately as potential outliers.

17. Card 17

    Question

    What are the 1.5 × IQR outlier fences?

    Answer

    Lower fence = Q1 − 1.5(IQR); upper fence = Q3 + 1.5(IQR). Values beyond them are flagged as potential outliers.

18. Card 18

    Question

    How should two quantitative distributions be compared?

    Answer

    Compare shape, center, variability, and unusual features in context, using the same measure or display basis.

19. Card 19

    Question

    What does a z-score of −1.8 mean?

    Answer

    The value is 1.8 standard deviations below the mean.

20. Card 20

    Question

    Every observation is converted from meters to centimeters by multiplying by 100. What happens to the mean and standard deviation?

    Answer

    Both are multiplied by 100.

21. Card 21

    Question

    What should an investigative question identify so the conclusion has a clear scope?

    Answer

    The variable or parameter of interest and the population to which the conclusion may apply.

22. Card 22

    Question

    What is a census?

    Answer

    A study that collects data from every member of the population.

23. Card 23

    Question

    What makes a study an experiment?

    Answer

    Researchers deliberately assign treatments to experimental units.

24. Card 24

    Question

    How do prospective and retrospective observational studies differ?

    Answer

    A prospective study follows units forward and gathers future data; a retrospective study uses data from the past.

25. Card 25

    Question

    What is a confounding variable in an observational study?

    Answer

    A variable associated with both the explanatory and response variables that offers an alternative explanation for their relationship.

26. Card 26

    Question

    What study feature supports generalizing results to a population?

    Answer

    Random selection from that population.

27. Card 27

    Question

    What makes a study observational?

    Answer

    Researchers observe variables without assigning treatments.

28. Card 28

    Question

    What study feature supports a cause-and-effect conclusion?

    Answer

    Random assignment of treatments in a well-designed experiment.

29. Card 29

    Question

    What defines a simple random sample of size n?

    Answer

    Every possible sample of size n has the same chance of selection.

30. Card 30

    Question

    What changes when sampling is done with replacement?

    Answer

    A selected unit returns to the population and can be selected again.

31. Card 31

    Question

    Why can a convenience sample be biased?

    Answer

    Easy-to-reach units may differ systematically from the target population.

32. Card 32

    Question

    Why should an experiment compare at least two treatment groups?

    Answer

    The comparison provides a baseline for judging whether responses differ by treatment.

33. Card 33

    Question

    A school samples 20 students at random from each grade. Which sampling method is this?

    Answer

    Stratified random sampling, with grade as the stratum.

34. Card 34

    Question

    What is the purpose of random assignment?

    Answer

    It tends to balance lurking variables across treatment groups, supporting causal inference.

35. Card 35

    Question

    Why can a voluntary-response sample be biased?

    Answer

    People with strong opinions are often more likely to participate.

36. Card 36

    Question

    What does replication mean in an experiment?

    Answer

    Assigning more than one experimental unit to each treatment so treatment differences can be separated from individual variability.

37. Card 37

    Question

    A city randomly selects 8 apartment buildings and surveys every household in those buildings. Which method is this?

    Answer

    Cluster random sampling.

38. Card 38

    Question

    What does direct control do in an experiment?

    Answer

    It holds potential extraneous sources of variation constant across experimental units.

39. Card 39

    Question

    What is undercoverage?

    Answer

    Some groups in the target population are left out of, or poorly represented in, the sampling frame.

40. Card 40

    Question

    What is the role of a control group?

    Answer

    It supplies a comparison condition for evaluating the treatment of interest.

41. Card 41

    Question

    After a random start, a quality inspector checks every 40th item. Which sampling method is this?

    Answer

    Systematic random sampling.

42. Card 42

    Question

    Why might an experiment use a placebo?

    Answer

    To separate a treatment's effect from responses caused by expecting treatment.

43. Card 43

    Question

    What is nonresponse bias?

    Answer

    Selected individuals who do not respond differ in a relevant way from those who do.

44. Card 44

    Question

    What is single blinding designed to reduce?

    Answer

    Bias caused when participants or evaluators know which treatment was received, depending on who is blinded.

45. Card 45

    Question

    Why use a randomized block design?

    Answer

    To group units that are similar on an important source of variation, then compare treatments within each block.

46. Card 46

    Question

    What defines a matched-pairs design?

    Answer

    Two treatments are compared using paired similar units or by giving both treatments to each unit in randomized order.

47. Card 47

    Question

    A survey asks, “Don't you agree the new schedule is unfair?” What problem does this create?

    Answer

    Response bias from leading wording.

48. Card 48

    Question

    What usually makes an experiment double-blind?

    Answer

    Neither the participants nor the people evaluating responses know treatment assignments while outcomes are measured.

49. Card 49

    Question

    A researcher randomly assigns 80 volunteers to two diets and compares blood-pressure change. What conclusion can random assignment support?

    Answer

    A cause-and-effect conclusion for people similar to the volunteers, assuming the experiment is well designed; volunteer recruitment does not support broad population generalization.

50. Card 50

    Question

    A researcher records coffee intake and sleep duration without assigning either. Can the study establish that coffee causes less sleep?

    Answer

    No. It is observational, so confounding can provide alternative explanations.

51. Card 51

    Question

    What is the difference between a population and a sample?

    Answer

    The population is the full group of interest; a sample is the subset actually observed.

52. Card 52

    Question

    Which graph is appropriate for the distribution of one quantitative variable measured on 600 people?

    Answer

    A histogram is appropriate; it groups the many numerical values into intervals.

53. Card 53

    Question

    In a strongly right-skewed distribution, how do the mean and median usually compare?

    Answer

    The mean is usually larger because high values pull it to the right.

54. Card 54

    Question

    Every score increases by 7 points. What happens to the mean and standard deviation?

    Answer

    The mean increases by 7; the standard deviation stays unchanged.

55. Card 55

    Question

    What does it mean that a score is at the 80th percentile?

    Answer

    About 80% of scores are at or below it.

56. Card 56

    Question

    Why should gaps and clusters be mentioned when describing a distribution?

    Answer

    They may reveal distinct subgroups, collection effects, or other structure that center and spread alone hide.

57. Card 57

    Question

    What is the minimum ethical safeguard when collecting identifiable human data?

    Answer

    Obtain informed consent when required and protect participants' privacy and confidentiality.

58. Card 58

    Question

    Every measurement is multiplied by −2. What happens to the mean and standard deviation?

    Answer

    The mean is multiplied by −2; the standard deviation is multiplied by 2.

59. Card 59

    Question

    A study uses random sampling but no assigned treatment. What can it support?

    Answer

    Population generalization, but not a cause-and-effect conclusion.

60. Card 60

    Question

    A report calls any unmeasured variable a confounder. What is the correction?

    Answer

    A confounder must be related to both the explanatory and response variables and create an alternative explanation.

61. Card 61

    Question

    What does a two-way table summarize?

    Answer

    Counts or relative frequencies for combinations of two categorical variables.

62. Card 62

    Question

    What is a joint relative frequency?

    Answer

    A cell count divided by the grand total, representing one combination of categories.

63. Card 63

    Question

    What is a marginal relative frequency?

    Answer

    A row or column total divided by the grand total.

64. Card 64

    Question

    How is a conditional relative frequency calculated within one row?

    Answer

    Divide each cell in that row by the row total.

65. Card 65

    Question

    What pattern suggests association between two categorical variables?

    Answer

    The conditional distribution of one variable changes across categories of the other.

66. Card 66

    Question

    Why are segmented bar charts useful for two categorical variables?

    Answer

    They place conditional distributions on the same 100% scale, making category patterns easy to compare.

67. Card 67

    Question

    How do an outcome and an event differ?

    Answer

    An outcome is one result of a trial; an event is a set of one or more outcomes.

68. Card 68

    Question

    What must a valid probability simulation specify?

    Answer

    A chance mechanism whose outcomes match the event probabilities, one trial definition, the statistic recorded, and many repetitions.

69. Card 69

    Question

    What does the law of large numbers predict?

    Answer

    As independent trials accumulate, an event's long-run relative frequency tends to approach its probability.

70. Card 70

    Question

    What two requirements must probabilities in a sample space satisfy?

    Answer

    Each probability is between 0 and 1, and the probabilities of all nonoverlapping outcomes sum to 1.

71. Card 71

    Question

    What is the complement rule?

    Answer

    P(Aᶜ) = 1 − P(A). It is often useful for “at least one” events.

72. Card 72

    Question

    How can you verify that events A and B are mutually exclusive?

    Answer

    Their intersection is impossible, so P(A ∩ B) = 0.

73. Card 73

    Question

    What is the formula for P(A | B), when P(B) > 0?

    Answer

    P(A | B) = P(A ∩ B) / P(B). The restricted sample space is B.

74. Card 74

    Question

    What is the general multiplication rule for two events?

    Answer

    P(A ∩ B) = P(A)P(B | A), or equivalently P(B)P(A | B).

75. Card 75

    Question

    What does it mean for events A and B to be independent?

    Answer

    Knowing that one occurred does not change the probability of the other.

76. Card 76

    Question

    What is the general addition rule?

    Answer

    P(A ∪ B) = P(A) + P(B) − P(A ∩ B).

77. Card 77

    Question

    Why are two mutually exclusive events with positive probabilities not independent?

    Answer

    If one occurs, the other cannot occur, so its conditional probability drops to 0.

78. Card 78

    Question

    What is a random variable?

    Answer

    A numerical value determined by the outcome of a random process.

79. Card 79

    Question

    What makes a table a valid discrete probability distribution?

    Answer

    It lists every possible value with probabilities from 0 to 1 that sum to 1.

80. Card 80

    Question

    What does a cumulative distribution value F(x) represent?

    Answer

    P(X ≤ x), the probability that the random variable is at most x.

81. Card 81

    Question

    How is the expected value of a discrete random variable calculated?

    Answer

    Multiply each possible value by its probability and add: E(X) = ΣxP(X = x).

82. Card 82

    Question

    What does the standard deviation of a random variable measure?

    Answer

    The typical distance of long-run outcomes from the random variable's mean.

83. Card 83

    Question

    How is the standard deviation of a discrete random variable calculated?

    Answer

    σₓ = √[Σ(x − μₓ)²P(X = x)]. The quantity inside the square root is Var(X).

84. Card 84

    Question

    A game has E(X) = −$0.40 per play. What does this mean?

    Answer

    Over many plays, the player's average net result approaches a loss of 40 cents per play; it does not predict every play.

85. Card 85

    Question

    What conditions define a binomial random variable?

    Answer

    A fixed number of independent trials, two outcomes per trial, constant success probability, and X counts successes.

86. Card 86

    Question

    For X ~ Binomial(n, p), what are the mean and standard deviation?

    Answer

    Mean = np; standard deviation = √[np(1 − p)].

87. Card 87

    Question

    For X ~ Binomial(n, p), what is P(X = x)?

    Answer

    Choose x success positions, then multiply: C(n, x)pˣ(1 − p)ⁿ⁻ˣ.

88. Card 88

    Question

    How can P(X ≥ 1) be found efficiently for a binomial variable?

    Answer

    Use the complement: P(X ≥ 1) = 1 − P(X = 0).

89. Card 89

    Question

    What should one simulated trial represent when estimating P(X ≥ 4) for X ~ Binomial(10, 0.3)?

    Answer

    Ten independent success/failure observations with success probability 0.3, followed by recording whether at least four successes occurred.

90. Card 90

    Question

    What features characterize a normal distribution?

    Answer

    It is continuous, symmetric, unimodal, and bell-shaped.

91. Card 91

    Question

    Which parameters determine a normal distribution?

    Answer

    Its mean μ sets the center, and its standard deviation σ sets the spread.

92. Card 92

    Question

    What is the standard normal distribution?

    Answer

    The normal distribution with mean 0 and standard deviation 1.

93. Card 93

    Question

    What is the 68–95–99.7 rule?

    Answer

    For an approximately normal distribution, about 68%, 95%, and 99.7% of values lie within 1, 2, and 3 standard deviations of the mean.

94. Card 94

    Question

    What does an area under a normal curve represent?

    Answer

    The probability or population proportion within the corresponding interval.

95. Card 95

    Question

    How do you find the value cutting off the lowest 10% of a normal distribution?

    Answer

    Find the z-score with cumulative area 0.10, then convert with x = μ + zσ.

96. Card 96

    Question

    A normal variable has μ = 50 and σ = 8. What z-score corresponds to x = 62?

    Answer

    1.5, because z = (62 − 50) / 8.

97. Card 97

    Question

    Two exam scores come from different normal distributions. What makes their percentiles comparable?

    Answer

    Standardize each score with its own distribution's mean and standard deviation, then compare z-scores or cumulative areas.

98. Card 98

    Question

    What is a sampling distribution of a statistic?

    Answer

    The distribution of that statistic over all possible random samples of a fixed size from a population.

99. Card 99

    Question

    How can a sampling distribution be approximated by simulation?

    Answer

    Repeatedly take random samples of the same size, calculate the statistic each time, and graph the resulting values.

100. Card 100

     Question

     What is a randomization distribution?

     Answer

     A simulated distribution of a statistic produced by repeatedly reallocating responses or labels as specified by a null model.

101. Card 101

     Question

     What does the central limit theorem say about sample means?

     Answer

     For random samples, the sampling distribution of the sample mean becomes approximately normal as sample size grows, even when the population is not normal.

102. Card 102

     Question

     How does increasing sample size affect the normal approximation in the central limit theorem?

     Answer

     It generally improves the approximation, especially for skewed or irregular populations.

103. Card 103

     Question

     A segmented bar chart shows nearly identical category proportions for every group. What does that suggest?

     Answer

     Little or no association between the two categorical variables.

104. Card 104

     Question

     In a survey, 30 of 120 students both bike to school and arrive before 8:00. What is the joint relative frequency?

     Answer

     0.25, because 30 / 120 = 0.25.

105. Card 105

     Question

     Why can P(A | B) differ from P(B | A)?

     Answer

     They use different restricted sample spaces and usually have different denominators.

106. Card 106

     Question

     If P(A) = 0.4 and P(A | B) = 0.4 with P(B) > 0, what does this indicate?

     Answer

     A and B are independent because learning B does not change the probability of A.

107. Card 107

     Question

     If independent events have probabilities 0.6 and 0.5, what is the probability that both occur?

     Answer

     0.30, using P(A ∩ B) = P(A)P(B).

108. Card 108

     Question

     A prize is $0 with probability 0.7 and $10 with probability 0.3. What is the expected prize?

     Answer

     $3, because 0(0.7) + 10(0.3) = 3.

109. Card 109

     Question

     A machine produces defective items independently with probability 0.02. What distribution models the number of defectives in 50 items?

     Answer

     Binomial with n = 50 and p = 0.02.

110. Card 110

     Question

     Heights are approximately normal with μ = 170 cm and σ = 6 cm. About what percent lie from 158 to 182 cm?

     Answer

     About 95%, because the interval is μ ± 2σ.

111. Card 111

     Question

     What makes an estimator unbiased?

     Answer

     Its sampling distribution is centered at the population parameter it estimates.

112. Card 112

     Question

     For random samples of size n, what is the mean of the sampling distribution of p̂?

     Answer

     μₚ̂ = p, where p is the population proportion.

113. Card 113

     Question

     Which procedure estimates one population proportion from a random sample?

     Answer

     A one-sample z-interval for a population proportion.

114. Card 114

     Question

     How should a confidence interval for a population proportion be interpreted?

     Answer

     We are confident at the stated level that the interval captures the true population proportion, in context.

115. Card 115

     Question

     What hypotheses test whether a population proportion differs from 0.40?

     Answer

     H₀: p = 0.40 versus Hₐ: p ≠ 0.40.

116. Card 116

     Question

     What is a p-value?

     Answer

     Assuming H₀ is true, it is the probability of a test statistic as extreme as or more extreme than the observed statistic in the direction of Hₐ.

117. Card 117

     Question

     What is the hypothesis-test decision rule using significance level α?

     Answer

     Reject H₀ when the p-value ≤ α; otherwise fail to reject H₀.

118. Card 118

     Question

     What is a Type I error?

     Answer

     Rejecting H₀ when H₀ is actually true.

119. Card 119

     Question

     What is the mean of p̂₁ − p̂₂ for independent random samples?

     Answer

     p₁ − p₂.

120. Card 120

     Question

     Which procedure estimates p₁ − p₂ from two independent samples or randomized groups?

     Answer

     A two-sample z-interval for a difference between population proportions.

121. Card 121

     Question

     How should a confidence interval for p₁ − p₂ be interpreted?

     Answer

     We are confident at the stated level that the interval captures the true difference p₁ − p₂, in context.

122. Card 122

     Question

     What null hypothesis is standard when testing whether two population proportions differ?

     Answer

     H₀: p₁ − p₂ = 0, equivalently p₁ = p₂.

123. Card 123

     Question

     A two-proportion test gives p-value 0.018 at α = 0.05. What decision follows?

     Answer

     Reject H₀ because 0.018 < 0.05.

124. Card 124

     Question

     When is a chi-square test for independence appropriate?

     Answer

     When one random sample provides two categorical variables and the question asks whether they are associated in one population.

125. Card 125

     Question

     How should a chi-square test p-value be interpreted?

     Answer

     Assuming the null model of independence or homogeneity is true, it is the probability of a chi-square statistic at least as large as the one observed.

     

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126. Card 126

     Question

     How do bias and variability differ for an estimator?

     Answer

     Bias concerns where the sampling distribution is centered; variability concerns how spread out it is.

127. Card 127

     Question

     What is the standard deviation of p̂ when observations are independent?

     Answer

     σₚ̂ = √[p(1 − p) / n].

128. Card 128

     Question

     What is the one-proportion z-interval formula?

     Answer

     p̂ ± z*√[p̂(1 − p̂) / n].

129. Card 129

     Question

     What does a 95% confidence level describe?

     Answer

     In repeated random sampling with the same method, about 95% of the resulting intervals would capture the true parameter.

130. Card 130

     Question

     Which method tests a claim about one population proportion when its conditions hold?

     Answer

     A one-sample z-test for a population proportion.

131. Card 131

     Question

     How does the alternative hypothesis determine a p-value's tail area?

     Answer

     A greater-than alternative uses the upper tail, a less-than alternative uses the lower tail, and a not-equal alternative uses both tails.

132. Card 132

     Question

     What wording should follow a rejected null hypothesis?

     Answer

     There is convincing statistical evidence for the alternative claim about the population parameter, stated in context.

133. Card 133

     Question

     What is a Type II error?

     Answer

     Failing to reject H₀ when Hₐ is actually true.

134. Card 134

     Question

     What is the standard deviation of p̂₁ − p̂₂ for independent samples?

     Answer

     √[p₁(1 − p₁)/n₁ + p₂(1 − p₂)/n₂].

135. Card 135

     Question

     What standard error is used in a confidence interval for p₁ − p₂?

     Answer

     √[p̂₁(1 − p̂₁)/n₁ + p̂₂(1 − p̂₂)/n₂]; the sample proportions are not pooled.

136. Card 136

     Question

     A confidence interval for p₁ − p₂ contains 0. What does that imply?

     Answer

     The interval does not provide convincing evidence of a difference between the population proportions at the corresponding two-sided significance level.

137. Card 137

     Question

     Why is a pooled proportion used in a two-proportion z-test with H₀: p₁ = p₂?

     Answer

     The null model assumes both samples share one common population proportion, estimated by combining successes and observations.

138. Card 138

     Question

     How should a p-value for a two-proportion test be stated?

     Answer

     Assuming the population proportions are equal, it is the probability of observing a difference in sample proportions at least as extreme as the one found, in the direction of Hₐ.

139. Card 139

     Question

     When is a chi-square test for homogeneity appropriate?

     Answer

     When independent samples or randomized groups are compared on the distribution of one categorical response variable.

140. Card 140

     Question

     What is the chi-square test statistic formula?

     Answer

     χ² = Σ[(observed − expected)² / expected], summed over all cells.

141. Card 141

     Question

     What usually happens to an estimator's sampling variability as sample size increases?

     Answer

     It decreases; estimates from larger random samples tend to cluster more tightly around the parameter.

142. Card 142

     Question

     When is the sampling distribution of p̂ approximately normal?

     Answer

     When the expected counts np and n(1 − p) are both at least 10.

143. Card 143

     Question

     What conditions justify a one-proportion z-interval?

     Answer

     Random data; independence, checked with n ≤ 10% of the population when sampling without replacement; and at least 10 observed successes and 10 observed failures.

144. Card 144

     Question

     A 95% confidence interval for p is (0.52, 0.61). What does it say about the claim p = 0.50?

     Answer

     The interval excludes 0.50, so the data provide evidence against p = 0.50 in a two-sided test at α = 0.05.

145. Card 145

     Question

     What is the one-proportion z-test statistic?

     Answer

     z = (p̂ − p₀) / √[p₀(1 − p₀)/n], using the null proportion p₀ in the standard error.

146. Card 146

     Question

     How is a simulation-based p-value estimated?

     Answer

     Find the proportion of simulated null statistics at least as extreme as the observed statistic in the direction of Hₐ.

147. Card 147

     Question

     What does “fail to reject H₀” mean?

     Answer

     The data do not provide convincing evidence for Hₐ; it does not prove H₀ true.

148. Card 148

     Question

     With sample size and effect fixed, what often happens when α is lowered?

     Answer

     The chance of a Type I error decreases, while the chance of a Type II error increases.

149. Card 149

     Question

     What conditions support the usual model for p̂₁ − p̂₂?

     Answer

     Independent random samples or randomized groups, independence within each group, and large enough expected success and failure counts for normal approximation.

150. Card 150

     Question

     What conditions justify a two-proportion z-interval?

     Answer

     Independent random samples or randomized groups; each sample no more than 10% of its population when sampling without replacement; and at least 10 observed successes and failures in each group.

151. Card 151

     Question

     How does increasing both sample sizes affect a confidence interval for p₁ − p₂?

     Answer

     It reduces the standard error and usually narrows the interval when other factors stay the same.

152. Card 152

     Question

     What standard error is used in the two-proportion z-test?

     Answer

     √[p̂c(1 − p̂c)(1/n₁ + 1/n₂)], where p̂c is the pooled sample proportion.

153. Card 153

     Question

     A randomized experiment uses volunteers assigned to two treatments. A significant two-proportion test supports what scope?

     Answer

     A cause-and-effect conclusion for people similar to the volunteers, not automatic generalization to a broader population.

154. Card 154

     Question

     How is an expected count computed in a two-way table under independence?

     Answer

     Expected count = (row total × column total) / grand total.

155. Card 155

     Question

     What conditions justify a chi-square test for a two-way table?

     Answer

     Random data; independent observations, including the 10% check when sampling without replacement; and every expected cell count greater than 5.

156. Card 156

     Question

     A sampling distribution is centered away from the true parameter. What problem does this reveal?

     Answer

     Bias in the estimator.

157. Card 157

     Question

     If p = 0.30 and n = 100, what does μₚ̂ = 0.30 mean?

     Answer

     Across many random samples of 100, the average sample proportion would be 0.30.

158. Card 158

     Question

     For a planned proportion interval with margin of error m, what conservative p-value is used when no prior estimate exists?

     Answer

     Use p* = 0.50 in n ≥ (z*/m)²p*(1 − p*) because it gives the largest required sample size.

159. Card 159

     Question

     What two changes widen a confidence interval for a proportion?

     Answer

     Using a higher confidence level or a smaller sample size.

160. Card 160

     Question

     Which counts check normality for a one-proportion z-test?

     Answer

     Use the null model: np₀ ≥ 10 and n(1 − p₀) ≥ 10.

161. Card 161

     Question

     What is wrong with saying “the p-value is the probability that H₀ is true”?

     Answer

     The p-value assumes H₀ is true and measures how unusual the observed statistic would be under that assumption; it does not assign probability to H₀.

162. Card 162

     Question

     What does “statistically significant at α = 0.01” mean?

     Answer

     The p-value is at most 0.01, so H₀ is rejected at that significance level.

163. Card 163

     Question

     What is the power of a hypothesis test?

     Answer

     The probability that the test rejects H₀ when a particular alternative is true.

164. Card 164

     Question

     If p₁ = p₂, where is the sampling distribution of p̂₁ − p̂₂ centered?

     Answer

     At 0, because its mean is p₁ − p₂.

165. Card 165

     Question

     Why must the order p̂₁ − p̂₂ stay consistent throughout an interval?

     Answer

     Changing the order reverses the sign and changes the contextual interpretation of every endpoint.

166. Card 166

     Question

     A 95% interval for p₁ − p₂ is (0.04, 0.15). What conclusion is supported?

     Answer

     p₁ is plausibly 0.04 to 0.15 higher than p₂; the interval supports a positive difference.

167. Card 167

     Question

     Which success-failure counts are checked for a two-proportion z-test?

     Answer

     Expected counts based on the pooled null proportion: n₁p̂c, n₁(1 − p̂c), n₂p̂c, and n₂(1 − p̂c), each at least 10.

168. Card 168

     Question

     A two-proportion test with Hₐ: p₁ ≠ p₂ fails to reject H₀. What conclusion is valid?

     Answer

     There is not convincing evidence that the two population proportions differ.

169. Card 169

     Question

     What are the degrees of freedom for a chi-square test on an r × c table?

     Answer

     (r − 1)(c − 1).

170. Card 170

     Question

     A chi-square test for independence has a small p-value. What conclusion is appropriate?

     Answer

     There is convincing evidence of an association between the two categorical variables in the population, stated in context.

171. Card 171

     Question

     What is the mean of the sampling distribution of x̄ for random samples from a population with mean μ?

     Answer

     μₓ̄ = μ.

172. Card 172

     Question

     Which procedure estimates one population mean when the population standard deviation is unknown?

     Answer

     A one-sample t-interval for a population mean.

173. Card 173

     Question

     How should a confidence interval for a population mean be interpreted?

     Answer

     We are confident at the stated level that the interval captures the true population mean, in context.

174. Card 174

     Question

     What hypotheses test whether a population mean exceeds 12?

     Answer

     H₀: μ = 12 versus Hₐ: μ > 12.

175. Card 175

     Question

     A one-sample t-test gives p-value 0.08 at α = 0.05. What decision follows?

     Answer

     Fail to reject H₀ because 0.08 > 0.05.

176. Card 176

     Question

     What is the mean of x̄₁ − x̄₂ for independent random samples?

     Answer

     μ₁ − μ₂.

177. Card 177

     Question

     Which procedure estimates μ₁ − μ₂ from two independent samples?

     Answer

     A two-sample t-interval for a difference between population means.

178. Card 178

     Question

     How should a confidence interval for μ₁ − μ₂ be interpreted?

     Answer

     We are confident at the stated level that the interval captures the true difference μ₁ − μ₂, in context.

179. Card 179

     Question

     What null hypothesis is standard when testing whether two population means differ?

     Answer

     H₀: μ₁ − μ₂ = 0, equivalently μ₁ = μ₂.

180. Card 180

     Question

     A two-sample t-test gives p-value 0.004 at α = 0.01. What decision follows?

     Answer

     Reject H₀ because 0.004 < 0.01.

181. Card 181

     Question

     What is the standard deviation of x̄ when observations are independent?

     Answer

     σₓ̄ = σ / √n.

182. Card 182

     Question

     What is the one-sample t-interval formula for μ?

     Answer

     x̄ ± t* × s/√n, with t* based on n − 1 degrees of freedom.

183. Card 183

     Question

     What does a 90% confidence level mean for a mean interval procedure?

     Answer

     Across many random samples using the same procedure, about 90% of the intervals would capture the true population mean.

184. Card 184

     Question

     Which procedure tests a claim about one population mean when σ is unknown?

     Answer

     A one-sample t-test for a population mean.

185. Card 185

     Question

     How should a one-mean test p-value be interpreted?

     Answer

     Assuming the null mean is true, it is the probability of a t-statistic as extreme as or more extreme than observed in the direction of Hₐ.

186. Card 186

     Question

     What is the standard deviation of x̄₁ − x̄₂ for independent samples?

     Answer

     √(σ₁²/n₁ + σ₂²/n₂).

187. Card 187

     Question

     What standard error is used in a two-sample t-interval for μ₁ − μ₂?

     Answer

     √(s₁²/n₁ + s₂²/n₂).

188. Card 188

     Question

     A confidence interval for μ₁ − μ₂ contains 0. What does that imply?

     Answer

     The interval does not provide convincing evidence of a difference between the population means at the corresponding two-sided significance level.

189. Card 189

     Question

     What is the two-sample t-statistic for testing H₀: μ₁ − μ₂ = 0?

     Answer

     t = [(x̄₁ − x̄₂) − 0] / √(s₁²/n₁ + s₂²/n₂).

190. Card 190

     Question

     How should a two-mean test p-value be interpreted?

     Answer

     Assuming the population means are equal, it is the probability of a sample-mean difference at least as extreme as observed, standardized in the direction of Hₐ.

191. Card 191

     Question

     When is the sampling distribution of x̄ approximately normal?

     Answer

     When the population is approximately normal or the random sample is large enough for the central limit theorem to apply.

192. Card 192

     Question

     What conditions justify a one-sample t-interval?

     Answer

     Random data; independence, checked with n ≤ 10% of the population when sampling without replacement; and for the Normal/Large Sample condition, n ≥ 30 is sufficient, while n < 30 requires sample data with no strong skewness or outliers.

193. Card 193

     Question

     How does increasing sample size affect a confidence interval for μ?

     Answer

     It lowers the standard error and usually narrows the interval when confidence level and variability stay comparable.

194. Card 194

     Question

     What is the one-sample t-test statistic?

     Answer

     t = (x̄ − μ₀) / (s/√n), with n − 1 degrees of freedom.

195. Card 195

     Question

     A t-test fails to reject H₀. What should the conclusion avoid?

     Answer

     Avoid saying H₀ is true; say the data do not provide convincing evidence for Hₐ.

196. Card 196

     Question

     When is x̄₁ − x̄₂ approximately normal?

     Answer

     When both populations are approximately normal or both independent random samples are large enough for normal approximations.

197. Card 197

     Question

     What conditions justify a two-sample t-interval?

     Answer

     Independent random samples or randomized groups; each sample no more than 10% of its population when sampling without replacement; and for the Normal/Large Sample condition, both sample sizes ≥ 30 are sufficient, while either sample below 30 requires sample data with no strong skewness or outliers.

198. Card 198

     Question

     A 95% interval for μ₁ − μ₂ is (−7.2, −1.4). What does it support?

     Answer

     μ₁ is plausibly 1.4 to 7.2 units lower than μ₂; the interval supports a negative difference.

199. Card 199

     Question

     What sample-shape condition is checked for a two-sample t-test with small samples?

     Answer

     Both sample distributions should be free of strong skewness and outliers unless both populations are known to be approximately normal.

200. Card 200

     Question

     A randomized experiment finds a significant difference in mean response. What can random assignment support?

     Answer

     A cause-and-effect conclusion for units like those studied, assuming the experiment was well designed.

201. Card 201

     Question

     A population has μ = 40. What does μₓ̄ = 40 mean for samples of size 25?

     Answer

     Across all random samples of 25, the average sample mean is 40.

202. Card 202

     Question

     How is a matched-pairs confidence interval analyzed?

     Answer

     Compute one difference for each pair, then use a one-sample t-interval on the population mean difference.

203. Card 203

     Question

     A 95% confidence interval for μ is (18.2, 21.7). What does it say about μ = 22?

     Answer

     The interval excludes 22, providing evidence against μ = 22 in a two-sided test at α = 0.05.

204. Card 204

     Question

     Which observations enter a matched-pairs t-test?

     Answer

     The within-pair differences, not the two original columns treated as independent samples.

205. Card 205

     Question

     A test reports p-value 0.032. At which common levels is it significant: 0.05 or 0.01?

     Answer

     Significant at 0.05, but not at 0.01.

206. Card 206

     Question

     If μ₁ − μ₂ = 5, where is the sampling distribution of x̄₁ − x̄₂ centered?

     Answer

     At 5.

207. Card 207

     Question

     Does the standard AP two-sample t procedure require equal population variances?

     Answer

     No. It uses separate sample variances in the standard error rather than pooling them.

208. Card 208

     Question

     What two changes usually widen a confidence interval for μ₁ − μ₂?

     Answer

     Higher confidence or smaller sample sizes.

209. Card 209

     Question

     Why must the order x̄₁ − x̄₂ match the order μ₁ − μ₂ in the hypotheses?

     Answer

     Reversing the order reverses the sign and changes the direction of the claim.

210. Card 210

     Question

     A two-sample test with Hₐ: μ₁ > μ₂ fails to reject H₀. What conclusion is valid?

     Answer

     There is not convincing evidence that μ₁ exceeds μ₂.

211. Card 211

     Question

     A population has σ = 18 and random samples have n = 36. What is σₓ̄?

     Answer

     3, because 18/√36 = 3.

212. Card 212

     Question

     Why is a t distribution used for inference about a mean when σ is unknown?

     Answer

     Replacing σ with the sample standard deviation s adds uncertainty, which the heavier-tailed t distribution accounts for.

213. Card 213

     Question

     What conditions justify a one-sample t-test?

     Answer

     Random data; independence, checked with n ≤ 10% of the population when sampling without replacement; and for the Normal/Large Sample condition, n ≥ 30 is sufficient, while n < 30 requires sample data with no strong skewness or outliers.

214. Card 214

     Question

     What distinguishes a two-sample means procedure from a matched-pairs procedure?

     Answer

     Two-sample procedures use independent groups; matched-pairs procedures analyze linked observations through their differences.

215. Card 215

     Question

     How are degrees of freedom handled for a two-sample t procedure?

     Answer

     Technology usually uses an approximation based on both sample variances and sizes; a conservative fallback uses the smaller of n₁ − 1 and n₂ − 1.

216. Card 216

     Question

     What type of variables belong on a scatterplot?

     Answer

     Two quantitative variables measured on the same observational units.

217. Card 217

     Question

     What does the correlation coefficient r describe?

     Answer

     The direction and strength of a linear relationship between two quantitative variables.

218. Card 218

     Question

     What does ŷ = a + bx represent?

     Answer

     A linear regression model predicting response y from explanatory variable x.

219. Card 219

     Question

     What is a residual?

     Answer

     Observed response minus predicted response: residual = y − ŷ.

220. Card 220

     Question

     What makes a regression line the least-squares line?

     Answer

     It minimizes the sum of squared residuals.

221. Card 221

     Question

     What four features should a scatterplot description address?

     Answer

     Direction, form, strength, and unusual features such as outliers or clusters.

222. Card 222

     Question

     What values can r take?

     Answer

     Any value from −1 to 1, inclusive.

223. Card 223

     Question

     How is the slope b interpreted in context?

     Answer

     For each one-unit increase in x, the predicted value of y changes by b units on average.

224. Card 224

     Question

     What does a positive residual mean?

     Answer

     The observed response is above the model's predicted response.

225. Card 225

     Question

     What is the least-squares slope formula?

     Answer

     b = r(sᵧ/sₓ).

226. Card 226

     Question

     A scatterplot trends downward from left to right. What direction is the association?

     Answer

     Negative: larger x-values tend to occur with smaller y-values.

227. Card 227

     Question

     Why can r be near 0 even when two variables are strongly related?

     Answer

     Correlation measures only linear association, so a strong curved relationship can have r near 0.

228. Card 228

     Question

     How is the intercept a interpreted in context?

     Answer

     It is the predicted response when x = 0, provided x = 0 is meaningful and within the data's scope.

229. Card 229

     Question

     A model predicts 18, and the observed response is 21. What is the residual?

     Answer

     3, because 21 − 18 = 3.

230. Card 230

     Question

     How is the least-squares intercept found from the slope?

     Answer

     a = ȳ − bx̄.

231. Card 231

     Question

     What makes a linear association look strong?

     Answer

     The points lie close to a straight-line pattern, regardless of whether the slope is steep or shallow.

232. Card 232

     Question

     Does r have measurement units?

     Answer

     No. Correlation is unitless because it is based on standardized values.

233. Card 233

     Question

     For ŷ = 12 + 2.5x, what is predicted when x = 4?

     Answer

     22, because 12 + 2.5(4) = 22.

234. Card 234

     Question

     What residual-plot pattern supports using a linear model?

     Answer

     Random scatter around zero with no clear curve, trend, or changing spread.

235. Card 235

     Question

     What does r² measure in simple linear regression?

     Answer

     The proportion of variation in the response variable explained by its linear relationship with the explanatory variable.

236. Card 236

     Question

     A scatterplot shows a strong association. Does that establish causation?

     Answer

     No. A scatterplot alone cannot rule out confounding or other explanations.

237. Card 237

     Question

     Why should unusual points be checked before interpreting r?

     Answer

     Correlation is not resistant; an outlier or influential point can change r substantially.

238. Card 238

     Question

     Why is extrapolation risky?

     Answer

     The relationship observed over the data range may not continue beyond that range.

239. Card 239

     Question

     A point lies below the regression line. What sign is its residual?

     Answer

     Negative, because observed y is less than predicted ŷ.

240. Card 240

     Question

     Which point always lies on a least-squares regression line with an intercept?

     Answer

     The point (x̄, ȳ).

241. Card 241

     Question

     Which variable goes on each axis of a scatterplot used for prediction?

     Answer

     The explanatory variable goes on the horizontal x-axis; the response variable goes on the vertical y-axis.

242. Card 242

     Question

     What happens to r if the roles of x and y are swapped?

     Answer

     Nothing. Correlation is symmetric.

243. Card 243

     Question

     What is interpolation?

     Answer

     Predicting a response for an x-value within the range of observed explanatory values.

244. Card 244

     Question

     A residual plot has a clear U-shape. What is the correction?

     Answer

     Do not treat the linear model as adequate; the curved pattern shows systematic structure remains.

245. Card 245

     Question

     A regression has r² = 0.64. What does this mean?

     Answer

     About 64% of the variation in the response is explained by its linear relationship with the explanatory variable.

246. Card 246

     Question

     What is an outlier in a scatterplot?

     Answer

     A point that falls away from the overall pattern of the other points.

247. Card 247

     Question

     What happens to r when x is converted from centimeters to meters?

     Answer

     It stays the same because multiplying by a positive constant does not change standardized linear association.

248. Card 248

     Question

     When can a regression relationship support a causal conclusion?

     Answer

     Only when the data come from a well-designed randomized experiment and the conclusion matches its scope.

249. Card 249

     Question

     What units does a residual use?

     Answer

     The same units as the response variable y.

250. Card 250

     Question

     What is an influential point in regression?

     Answer

     A point whose removal substantially changes the fitted regression line or another key regression result.

250 cards

AP Statistics Flashcards: Complete 5-Unit Course Review

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