AP Statistics Flashcards: Complete 5-Unit Course Review

这套卡组中的卡片

1. 卡片 1

   问题

   What makes a question a statistical investigative question?

   答案

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

2. 卡片 2

   问题

   What is an observational unit?

   答案

   An individual item or person from which data are collected.

3. 卡片 3

   问题

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

   答案

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

4. 卡片 4

   问题

   How does a parameter differ from a statistic?

   答案

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

5. 卡片 5

   问题

   How is a category's relative frequency calculated?

   答案

   Divide the category count by the total number of observations.

6. 卡片 6

   问题

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

   答案

   The proportion or percentage of observations in that category.

7. 卡片 7

   问题

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

   答案

   Discrete. It is a count with separated possible values.

8. 卡片 8

   问题

   Which displays preserve individual quantitative data values?

   答案

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

9. 卡片 9

   问题

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

   答案

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

10. 卡片 10

    问题

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

    答案

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

11. 卡片 11

    问题

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

    答案

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

12. 卡片 12

    问题

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

    答案

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

13. 卡片 13

    问题

    How is the interquartile range calculated?

    答案

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

14. 卡片 14

    问题

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

    答案

    Values typically lie close to the mean.

15. 卡片 15

    问题

    Which common summaries are resistant to extreme values?

    答案

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

16. 卡片 16

    问题

    In a modified boxplot, where do the whiskers end?

    答案

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

17. 卡片 17

    问题

    What are the 1.5 × IQR outlier fences?

    答案

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

18. 卡片 18

    问题

    How should two quantitative distributions be compared?

    答案

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

19. 卡片 19

    问题

    What does a z-score of −1.8 mean?

    答案

    The value is 1.8 standard deviations below the mean.

20. 卡片 20

    问题

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

    答案

    Both are multiplied by 100.

21. 卡片 21

    问题

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

    答案

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

22. 卡片 22

    问题

    What is a census?

    答案

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

23. 卡片 23

    问题

    What makes a study an experiment?

    答案

    Researchers deliberately assign treatments to experimental units.

24. 卡片 24

    问题

    How do prospective and retrospective observational studies differ?

    答案

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

25. 卡片 25

    问题

    What is a confounding variable in an observational study?

    答案

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

26. 卡片 26

    问题

    What study feature supports generalizing results to a population?

    答案

    Random selection from that population.

27. 卡片 27

    问题

    What makes a study observational?

    答案

    Researchers observe variables without assigning treatments.

28. 卡片 28

    问题

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

    答案

    Random assignment of treatments in a well-designed experiment.

29. 卡片 29

    问题

    What defines a simple random sample of size n?

    答案

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

30. 卡片 30

    问题

    What changes when sampling is done with replacement?

    答案

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

31. 卡片 31

    问题

    Why can a convenience sample be biased?

    答案

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

32. 卡片 32

    问题

    Why should an experiment compare at least two treatment groups?

    答案

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

33. 卡片 33

    问题

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

    答案

    Stratified random sampling, with grade as the stratum.

34. 卡片 34

    问题

    What is the purpose of random assignment?

    答案

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

35. 卡片 35

    问题

    Why can a voluntary-response sample be biased?

    答案

    People with strong opinions are often more likely to participate.

36. 卡片 36

    问题

    What does replication mean in an experiment?

    答案

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

37. 卡片 37

    问题

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

    答案

    Cluster random sampling.

38. 卡片 38

    问题

    What does direct control do in an experiment?

    答案

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

39. 卡片 39

    问题

    What is undercoverage?

    答案

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

40. 卡片 40

    问题

    What is the role of a control group?

    答案

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

41. 卡片 41

    问题

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

    答案

    Systematic random sampling.

42. 卡片 42

    问题

    Why might an experiment use a placebo?

    答案

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

43. 卡片 43

    问题

    What is nonresponse bias?

    答案

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

44. 卡片 44

    问题

    What is single blinding designed to reduce?

    答案

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

45. 卡片 45

    问题

    Why use a randomized block design?

    答案

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

46. 卡片 46

    问题

    What defines a matched-pairs design?

    答案

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

47. 卡片 47

    问题

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

    答案

    Response bias from leading wording.

48. 卡片 48

    问题

    What usually makes an experiment double-blind?

    答案

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

49. 卡片 49

    问题

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

    答案

    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. 卡片 50

    问题

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

    答案

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

51. 卡片 51

    问题

    What is the difference between a population and a sample?

    答案

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

52. 卡片 52

    问题

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

    答案

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

53. 卡片 53

    问题

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

    答案

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

54. 卡片 54

    问题

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

    答案

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

55. 卡片 55

    问题

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

    答案

    About 80% of scores are at or below it.

56. 卡片 56

    问题

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

    答案

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

57. 卡片 57

    问题

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

    答案

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

58. 卡片 58

    问题

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

    答案

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

59. 卡片 59

    问题

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

    答案

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

60. 卡片 60

    问题

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

    答案

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

61. 卡片 61

    问题

    What does a two-way table summarize?

    答案

    Counts or relative frequencies for combinations of two categorical variables.

62. 卡片 62

    问题

    What is a joint relative frequency?

    答案

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

63. 卡片 63

    问题

    What is a marginal relative frequency?

    答案

    A row or column total divided by the grand total.

64. 卡片 64

    问题

    How is a conditional relative frequency calculated within one row?

    答案

    Divide each cell in that row by the row total.

65. 卡片 65

    问题

    What pattern suggests association between two categorical variables?

    答案

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

66. 卡片 66

    问题

    Why are segmented bar charts useful for two categorical variables?

    答案

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

67. 卡片 67

    问题

    How do an outcome and an event differ?

    答案

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

68. 卡片 68

    问题

    What must a valid probability simulation specify?

    答案

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

69. 卡片 69

    问题

    What does the law of large numbers predict?

    答案

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

70. 卡片 70

    问题

    What two requirements must probabilities in a sample space satisfy?

    答案

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

71. 卡片 71

    问题

    What is the complement rule?

    答案

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

72. 卡片 72

    问题

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

    答案

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

73. 卡片 73

    问题

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

    答案

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

74. 卡片 74

    问题

    What is the general multiplication rule for two events?

    答案

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

75. 卡片 75

    问题

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

    答案

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

76. 卡片 76

    问题

    What is the general addition rule?

    答案

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

77. 卡片 77

    问题

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

    答案

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

78. 卡片 78

    问题

    What is a random variable?

    答案

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

79. 卡片 79

    问题

    What makes a table a valid discrete probability distribution?

    答案

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

80. 卡片 80

    问题

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

    答案

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

81. 卡片 81

    问题

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

    答案

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

82. 卡片 82

    问题

    What does the standard deviation of a random variable measure?

    答案

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

83. 卡片 83

    问题

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

    答案

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

84. 卡片 84

    问题

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

    答案

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

85. 卡片 85

    问题

    What conditions define a binomial random variable?

    答案

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

86. 卡片 86

    问题

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

    答案

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

87. 卡片 87

    问题

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

    答案

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

88. 卡片 88

    问题

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

    答案

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

89. 卡片 89

    问题

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

    答案

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

90. 卡片 90

    问题

    What features characterize a normal distribution?

    答案

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

91. 卡片 91

    问题

    Which parameters determine a normal distribution?

    答案

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

92. 卡片 92

    问题

    What is the standard normal distribution?

    答案

    The normal distribution with mean 0 and standard deviation 1.

93. 卡片 93

    问题

    What is the 68–95–99.7 rule?

    答案

    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. 卡片 94

    问题

    What does an area under a normal curve represent?

    答案

    The probability or population proportion within the corresponding interval.

95. 卡片 95

    问题

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

    答案

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

96. 卡片 96

    问题

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

    答案

    1.5, because z = (62 − 50) / 8.

97. 卡片 97

    问题

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

    答案

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

98. 卡片 98

    问题

    What is a sampling distribution of a statistic?

    答案

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

99. 卡片 99

    问题

    How can a sampling distribution be approximated by simulation?

    答案

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

100. 卡片 100

     问题

     What is a randomization distribution?

     答案

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

101. 卡片 101

     问题

     What does the central limit theorem say about sample means?

     答案

     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. 卡片 102

     问题

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

     答案

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

103. 卡片 103

     问题

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

     答案

     Little or no association between the two categorical variables.

104. 卡片 104

     问题

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

     答案

     0.25, because 30 / 120 = 0.25.

105. 卡片 105

     问题

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

     答案

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

106. 卡片 106

     问题

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

     答案

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

107. 卡片 107

     问题

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

     答案

     0.30, using P(A ∩ B) = P(A)P(B).

108. 卡片 108

     问题

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

     答案

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

109. 卡片 109

     问题

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

     答案

     Binomial with n = 50 and p = 0.02.

110. 卡片 110

     问题

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

     答案

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

111. 卡片 111

     问题

     What makes an estimator unbiased?

     答案

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

112. 卡片 112

     问题

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

     答案

     μₚ̂ = p, where p is the population proportion.

113. 卡片 113

     问题

     Which procedure estimates one population proportion from a random sample?

     答案

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

114. 卡片 114

     问题

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

     答案

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

115. 卡片 115

     问题

     What hypotheses test whether a population proportion differs from 0.40?

     答案

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

116. 卡片 116

     问题

     What is a p-value?

     答案

     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. 卡片 117

     问题

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

     答案

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

118. 卡片 118

     问题

     What is a Type I error?

     答案

     Rejecting H₀ when H₀ is actually true.

119. 卡片 119

     问题

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

     答案

     p₁ − p₂.

120. 卡片 120

     问题

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

     答案

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

121. 卡片 121

     问题

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

     答案

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

122. 卡片 122

     问题

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

     答案

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

123. 卡片 123

     问题

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

     答案

     Reject H₀ because 0.018 < 0.05.

124. 卡片 124

     问题

     When is a chi-square test for independence appropriate?

     答案

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

125. 卡片 125

     问题

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

     答案

     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. 卡片 126

     问题

     How do bias and variability differ for an estimator?

     答案

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

127. 卡片 127

     问题

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

     答案

     σₚ̂ = √[p(1 − p) / n].

128. 卡片 128

     问题

     What is the one-proportion z-interval formula?

     答案

     p̂ ± z*√[p̂(1 − p̂) / n].

129. 卡片 129

     问题

     What does a 95% confidence level describe?

     答案

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

130. 卡片 130

     问题

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

     答案

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

131. 卡片 131

     问题

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

     答案

     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. 卡片 132

     问题

     What wording should follow a rejected null hypothesis?

     答案

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

133. 卡片 133

     问题

     What is a Type II error?

     答案

     Failing to reject H₀ when Hₐ is actually true.

134. 卡片 134

     问题

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

     答案

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

135. 卡片 135

     问题

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

     答案

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

136. 卡片 136

     问题

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

     答案

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

137. 卡片 137

     问题

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

     答案

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

138. 卡片 138

     问题

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

     答案

     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. 卡片 139

     问题

     When is a chi-square test for homogeneity appropriate?

     答案

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

140. 卡片 140

     问题

     What is the chi-square test statistic formula?

     答案

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

141. 卡片 141

     问题

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

     答案

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

142. 卡片 142

     问题

     When is the sampling distribution of p̂ approximately normal?

     答案

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

143. 卡片 143

     问题

     What conditions justify a one-proportion z-interval?

     答案

     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. 卡片 144

     问题

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

     答案

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

145. 卡片 145

     问题

     What is the one-proportion z-test statistic?

     答案

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

146. 卡片 146

     问题

     How is a simulation-based p-value estimated?

     答案

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

147. 卡片 147

     问题

     What does “fail to reject H₀” mean?

     答案

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

148. 卡片 148

     问题

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

     答案

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

149. 卡片 149

     问题

     What conditions support the usual model for p̂₁ − p̂₂?

     答案

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

150. 卡片 150

     问题

     What conditions justify a two-proportion z-interval?

     答案

     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. 卡片 151

     问题

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

     答案

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

152. 卡片 152

     问题

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

     答案

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

153. 卡片 153

     问题

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

     答案

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

154. 卡片 154

     问题

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

     答案

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

155. 卡片 155

     问题

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

     答案

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

156. 卡片 156

     问题

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

     答案

     Bias in the estimator.

157. 卡片 157

     问题

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

     答案

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

158. 卡片 158

     问题

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

     答案

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

159. 卡片 159

     问题

     What two changes widen a confidence interval for a proportion?

     答案

     Using a higher confidence level or a smaller sample size.

160. 卡片 160

     问题

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

     答案

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

161. 卡片 161

     问题

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

     答案

     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. 卡片 162

     问题

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

     答案

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

163. 卡片 163

     问题

     What is the power of a hypothesis test?

     答案

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

164. 卡片 164

     问题

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

     答案

     At 0, because its mean is p₁ − p₂.

165. 卡片 165

     问题

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

     答案

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

166. 卡片 166

     问题

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

     答案

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

167. 卡片 167

     问题

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

     答案

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

168. 卡片 168

     问题

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

     答案

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

169. 卡片 169

     问题

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

     答案

     (r − 1)(c − 1).

170. 卡片 170

     问题

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

     答案

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

171. 卡片 171

     问题

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

     答案

     μₓ̄ = μ.

172. 卡片 172

     问题

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

     答案

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

173. 卡片 173

     问题

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

     答案

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

174. 卡片 174

     问题

     What hypotheses test whether a population mean exceeds 12?

     答案

     H₀: μ = 12 versus Hₐ: μ > 12.

175. 卡片 175

     问题

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

     答案

     Fail to reject H₀ because 0.08 > 0.05.

176. 卡片 176

     问题

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

     答案

     μ₁ − μ₂.

177. 卡片 177

     问题

     Which procedure estimates μ₁ − μ₂ from two independent samples?

     答案

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

178. 卡片 178

     问题

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

     答案

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

179. 卡片 179

     问题

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

     答案

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

180. 卡片 180

     问题

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

     答案

     Reject H₀ because 0.004 < 0.01.

181. 卡片 181

     问题

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

     答案

     σₓ̄ = σ / √n.

182. 卡片 182

     问题

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

     答案

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

183. 卡片 183

     问题

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

     答案

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

184. 卡片 184

     问题

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

     答案

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

185. 卡片 185

     问题

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

     答案

     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. 卡片 186

     问题

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

     答案

     √(σ₁²/n₁ + σ₂²/n₂).

187. 卡片 187

     问题

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

     答案

     √(s₁²/n₁ + s₂²/n₂).

188. 卡片 188

     问题

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

     答案

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

189. 卡片 189

     问题

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

     答案

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

190. 卡片 190

     问题

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

     答案

     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. 卡片 191

     问题

     When is the sampling distribution of x̄ approximately normal?

     答案

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

192. 卡片 192

     问题

     What conditions justify a one-sample t-interval?

     答案

     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. 卡片 193

     问题

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

     答案

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

194. 卡片 194

     问题

     What is the one-sample t-test statistic?

     答案

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

195. 卡片 195

     问题

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

     答案

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

196. 卡片 196

     问题

     When is x̄₁ − x̄₂ approximately normal?

     答案

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

197. 卡片 197

     问题

     What conditions justify a two-sample t-interval?

     答案

     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. 卡片 198

     问题

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

     答案

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

199. 卡片 199

     问题

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

     答案

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

200. 卡片 200

     问题

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

     答案

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

201. 卡片 201

     问题

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

     答案

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

202. 卡片 202

     问题

     How is a matched-pairs confidence interval analyzed?

     答案

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

203. 卡片 203

     问题

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

     答案

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

204. 卡片 204

     问题

     Which observations enter a matched-pairs t-test?

     答案

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

205. 卡片 205

     问题

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

     答案

     Significant at 0.05, but not at 0.01.

206. 卡片 206

     问题

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

     答案

     At 5.

207. 卡片 207

     问题

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

     答案

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

208. 卡片 208

     问题

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

     答案

     Higher confidence or smaller sample sizes.

209. 卡片 209

     问题

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

     答案

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

210. 卡片 210

     问题

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

     答案

     There is not convincing evidence that μ₁ exceeds μ₂.

211. 卡片 211

     问题

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

     答案

     3, because 18/√36 = 3.

212. 卡片 212

     问题

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

     答案

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

213. 卡片 213

     问题

     What conditions justify a one-sample t-test?

     答案

     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. 卡片 214

     问题

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

     答案

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

215. 卡片 215

     问题

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

     答案

     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. 卡片 216

     问题

     What type of variables belong on a scatterplot?

     答案

     Two quantitative variables measured on the same observational units.

217. 卡片 217

     问题

     What does the correlation coefficient r describe?

     答案

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

218. 卡片 218

     问题

     What does ŷ = a + bx represent?

     答案

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

219. 卡片 219

     问题

     What is a residual?

     答案

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

220. 卡片 220

     问题

     What makes a regression line the least-squares line?

     答案

     It minimizes the sum of squared residuals.

221. 卡片 221

     问题

     What four features should a scatterplot description address?

     答案

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

222. 卡片 222

     问题

     What values can r take?

     答案

     Any value from −1 to 1, inclusive.

223. 卡片 223

     问题

     How is the slope b interpreted in context?

     答案

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

224. 卡片 224

     问题

     What does a positive residual mean?

     答案

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

225. 卡片 225

     问题

     What is the least-squares slope formula?

     答案

     b = r(sᵧ/sₓ).

226. 卡片 226

     问题

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

     答案

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

227. 卡片 227

     问题

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

     答案

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

228. 卡片 228

     问题

     How is the intercept a interpreted in context?

     答案

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

229. 卡片 229

     问题

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

     答案

     3, because 21 − 18 = 3.

230. 卡片 230

     问题

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

     答案

     a = ȳ − bx̄.

231. 卡片 231

     问题

     What makes a linear association look strong?

     答案

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

232. 卡片 232

     问题

     Does r have measurement units?

     答案

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

233. 卡片 233

     问题

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

     答案

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

234. 卡片 234

     问题

     What residual-plot pattern supports using a linear model?

     答案

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

235. 卡片 235

     问题

     What does r² measure in simple linear regression?

     答案

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

236. 卡片 236

     问题

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

     答案

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

237. 卡片 237

     问题

     Why should unusual points be checked before interpreting r?

     答案

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

238. 卡片 238

     问题

     Why is extrapolation risky?

     答案

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

239. 卡片 239

     问题

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

     答案

     Negative, because observed y is less than predicted ŷ.

240. 卡片 240

     问题

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

     答案

     The point (x̄, ȳ).

241. 卡片 241

     问题

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

     答案

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

242. 卡片 242

     问题

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

     答案

     Nothing. Correlation is symmetric.

243. 卡片 243

     问题

     What is interpolation?

     答案

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

244. 卡片 244

     问题

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

     答案

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

245. 卡片 245

     问题

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

     答案

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

246. 卡片 246

     问题

     What is an outlier in a scatterplot?

     答案

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

247. 卡片 247

     问题

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

     答案

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

248. 卡片 248

     问题

     When can a regression relationship support a causal conclusion?

     答案

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

249. 卡片 249

     问题

     What units does a residual use?

     答案

     The same units as the response variable y.

250. 卡片 250

     问题

     What is an influential point in regression?

     答案

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

250张卡片

AP Statistics Flashcards: Complete 5-Unit Course Review

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