AP Statistics Flashcards: Complete 5-Unit Course Review

Cartes de ce paquet

1. Carte 1

   Question

   What makes a question a statistical investigative question?

   Réponse

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

2. Carte 2

   Question

   What is an observational unit?

   Réponse

   An individual item or person from which data are collected.

3. Carte 3

   Question

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

   Réponse

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

4. Carte 4

   Question

   How does a parameter differ from a statistic?

   Réponse

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

5. Carte 5

   Question

   How is a category's relative frequency calculated?

   Réponse

   Divide the category count by the total number of observations.

6. Carte 6

   Question

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

   Réponse

   The proportion or percentage of observations in that category.

7. Carte 7

   Question

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

   Réponse

   Discrete. It is a count with separated possible values.

8. Carte 8

   Question

   Which displays preserve individual quantitative data values?

   Réponse

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

9. Carte 9

   Question

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

   Réponse

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

10. Carte 10

    Question

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

    Réponse

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

11. Carte 11

    Question

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

    Réponse

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

12. Carte 12

    Question

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

    Réponse

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

13. Carte 13

    Question

    How is the interquartile range calculated?

    Réponse

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

14. Carte 14

    Question

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

    Réponse

    Values typically lie close to the mean.

15. Carte 15

    Question

    Which common summaries are resistant to extreme values?

    Réponse

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

16. Carte 16

    Question

    In a modified boxplot, where do the whiskers end?

    Réponse

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

17. Carte 17

    Question

    What are the 1.5 × IQR outlier fences?

    Réponse

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

18. Carte 18

    Question

    How should two quantitative distributions be compared?

    Réponse

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

19. Carte 19

    Question

    What does a z-score of −1.8 mean?

    Réponse

    The value is 1.8 standard deviations below the mean.

20. Carte 20

    Question

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

    Réponse

    Both are multiplied by 100.

21. Carte 21

    Question

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

    Réponse

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

22. Carte 22

    Question

    What is a census?

    Réponse

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

23. Carte 23

    Question

    What makes a study an experiment?

    Réponse

    Researchers deliberately assign treatments to experimental units.

24. Carte 24

    Question

    How do prospective and retrospective observational studies differ?

    Réponse

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

25. Carte 25

    Question

    What is a confounding variable in an observational study?

    Réponse

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

26. Carte 26

    Question

    What study feature supports generalizing results to a population?

    Réponse

    Random selection from that population.

27. Carte 27

    Question

    What makes a study observational?

    Réponse

    Researchers observe variables without assigning treatments.

28. Carte 28

    Question

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

    Réponse

    Random assignment of treatments in a well-designed experiment.

29. Carte 29

    Question

    What defines a simple random sample of size n?

    Réponse

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

30. Carte 30

    Question

    What changes when sampling is done with replacement?

    Réponse

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

31. Carte 31

    Question

    Why can a convenience sample be biased?

    Réponse

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

32. Carte 32

    Question

    Why should an experiment compare at least two treatment groups?

    Réponse

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

33. Carte 33

    Question

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

    Réponse

    Stratified random sampling, with grade as the stratum.

34. Carte 34

    Question

    What is the purpose of random assignment?

    Réponse

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

35. Carte 35

    Question

    Why can a voluntary-response sample be biased?

    Réponse

    People with strong opinions are often more likely to participate.

36. Carte 36

    Question

    What does replication mean in an experiment?

    Réponse

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

37. Carte 37

    Question

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

    Réponse

    Cluster random sampling.

38. Carte 38

    Question

    What does direct control do in an experiment?

    Réponse

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

39. Carte 39

    Question

    What is undercoverage?

    Réponse

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

40. Carte 40

    Question

    What is the role of a control group?

    Réponse

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

41. Carte 41

    Question

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

    Réponse

    Systematic random sampling.

42. Carte 42

    Question

    Why might an experiment use a placebo?

    Réponse

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

43. Carte 43

    Question

    What is nonresponse bias?

    Réponse

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

44. Carte 44

    Question

    What is single blinding designed to reduce?

    Réponse

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

45. Carte 45

    Question

    Why use a randomized block design?

    Réponse

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

46. Carte 46

    Question

    What defines a matched-pairs design?

    Réponse

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

47. Carte 47

    Question

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

    Réponse

    Response bias from leading wording.

48. Carte 48

    Question

    What usually makes an experiment double-blind?

    Réponse

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

49. Carte 49

    Question

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

    Réponse

    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. Carte 50

    Question

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

    Réponse

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

51. Carte 51

    Question

    What is the difference between a population and a sample?

    Réponse

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

52. Carte 52

    Question

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

    Réponse

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

53. Carte 53

    Question

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

    Réponse

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

54. Carte 54

    Question

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

    Réponse

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

55. Carte 55

    Question

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

    Réponse

    About 80% of scores are at or below it.

56. Carte 56

    Question

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

    Réponse

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

57. Carte 57

    Question

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

    Réponse

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

58. Carte 58

    Question

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

    Réponse

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

59. Carte 59

    Question

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

    Réponse

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

60. Carte 60

    Question

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

    Réponse

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

61. Carte 61

    Question

    What does a two-way table summarize?

    Réponse

    Counts or relative frequencies for combinations of two categorical variables.

62. Carte 62

    Question

    What is a joint relative frequency?

    Réponse

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

63. Carte 63

    Question

    What is a marginal relative frequency?

    Réponse

    A row or column total divided by the grand total.

64. Carte 64

    Question

    How is a conditional relative frequency calculated within one row?

    Réponse

    Divide each cell in that row by the row total.

65. Carte 65

    Question

    What pattern suggests association between two categorical variables?

    Réponse

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

66. Carte 66

    Question

    Why are segmented bar charts useful for two categorical variables?

    Réponse

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

67. Carte 67

    Question

    How do an outcome and an event differ?

    Réponse

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

68. Carte 68

    Question

    What must a valid probability simulation specify?

    Réponse

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

69. Carte 69

    Question

    What does the law of large numbers predict?

    Réponse

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

70. Carte 70

    Question

    What two requirements must probabilities in a sample space satisfy?

    Réponse

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

71. Carte 71

    Question

    What is the complement rule?

    Réponse

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

72. Carte 72

    Question

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

    Réponse

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

73. Carte 73

    Question

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

    Réponse

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

74. Carte 74

    Question

    What is the general multiplication rule for two events?

    Réponse

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

75. Carte 75

    Question

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

    Réponse

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

76. Carte 76

    Question

    What is the general addition rule?

    Réponse

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

77. Carte 77

    Question

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

    Réponse

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

78. Carte 78

    Question

    What is a random variable?

    Réponse

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

79. Carte 79

    Question

    What makes a table a valid discrete probability distribution?

    Réponse

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

80. Carte 80

    Question

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

    Réponse

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

81. Carte 81

    Question

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

    Réponse

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

82. Carte 82

    Question

    What does the standard deviation of a random variable measure?

    Réponse

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

83. Carte 83

    Question

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

    Réponse

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

84. Carte 84

    Question

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

    Réponse

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

85. Carte 85

    Question

    What conditions define a binomial random variable?

    Réponse

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

86. Carte 86

    Question

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

    Réponse

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

87. Carte 87

    Question

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

    Réponse

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

88. Carte 88

    Question

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

    Réponse

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

89. Carte 89

    Question

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

    Réponse

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

90. Carte 90

    Question

    What features characterize a normal distribution?

    Réponse

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

91. Carte 91

    Question

    Which parameters determine a normal distribution?

    Réponse

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

92. Carte 92

    Question

    What is the standard normal distribution?

    Réponse

    The normal distribution with mean 0 and standard deviation 1.

93. Carte 93

    Question

    What is the 68–95–99.7 rule?

    Réponse

    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. Carte 94

    Question

    What does an area under a normal curve represent?

    Réponse

    The probability or population proportion within the corresponding interval.

95. Carte 95

    Question

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

    Réponse

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

96. Carte 96

    Question

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

    Réponse

    1.5, because z = (62 − 50) / 8.

97. Carte 97

    Question

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

    Réponse

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

98. Carte 98

    Question

    What is a sampling distribution of a statistic?

    Réponse

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

99. Carte 99

    Question

    How can a sampling distribution be approximated by simulation?

    Réponse

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

100. Carte 100

     Question

     What is a randomization distribution?

     Réponse

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

101. Carte 101

     Question

     What does the central limit theorem say about sample means?

     Réponse

     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. Carte 102

     Question

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

     Réponse

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

103. Carte 103

     Question

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

     Réponse

     Little or no association between the two categorical variables.

104. Carte 104

     Question

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

     Réponse

     0.25, because 30 / 120 = 0.25.

105. Carte 105

     Question

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

     Réponse

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

106. Carte 106

     Question

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

     Réponse

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

107. Carte 107

     Question

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

     Réponse

     0.30, using P(A ∩ B) = P(A)P(B).

108. Carte 108

     Question

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

     Réponse

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

109. Carte 109

     Question

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

     Réponse

     Binomial with n = 50 and p = 0.02.

110. Carte 110

     Question

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

     Réponse

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

111. Carte 111

     Question

     What makes an estimator unbiased?

     Réponse

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

112. Carte 112

     Question

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

     Réponse

     μₚ̂ = p, where p is the population proportion.

113. Carte 113

     Question

     Which procedure estimates one population proportion from a random sample?

     Réponse

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

114. Carte 114

     Question

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

     Réponse

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

115. Carte 115

     Question

     What hypotheses test whether a population proportion differs from 0.40?

     Réponse

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

116. Carte 116

     Question

     What is a p-value?

     Réponse

     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. Carte 117

     Question

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

     Réponse

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

118. Carte 118

     Question

     What is a Type I error?

     Réponse

     Rejecting H₀ when H₀ is actually true.

119. Carte 119

     Question

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

     Réponse

     p₁ − p₂.

120. Carte 120

     Question

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

     Réponse

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

121. Carte 121

     Question

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

     Réponse

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

122. Carte 122

     Question

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

     Réponse

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

123. Carte 123

     Question

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

     Réponse

     Reject H₀ because 0.018 < 0.05.

124. Carte 124

     Question

     When is a chi-square test for independence appropriate?

     Réponse

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

125. Carte 125

     Question

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

     Réponse

     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. Carte 126

     Question

     How do bias and variability differ for an estimator?

     Réponse

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

127. Carte 127

     Question

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

     Réponse

     σₚ̂ = √[p(1 − p) / n].

128. Carte 128

     Question

     What is the one-proportion z-interval formula?

     Réponse

     p̂ ± z*√[p̂(1 − p̂) / n].

129. Carte 129

     Question

     What does a 95% confidence level describe?

     Réponse

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

130. Carte 130

     Question

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

     Réponse

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

131. Carte 131

     Question

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

     Réponse

     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. Carte 132

     Question

     What wording should follow a rejected null hypothesis?

     Réponse

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

133. Carte 133

     Question

     What is a Type II error?

     Réponse

     Failing to reject H₀ when Hₐ is actually true.

134. Carte 134

     Question

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

     Réponse

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

135. Carte 135

     Question

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

     Réponse

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

136. Carte 136

     Question

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

     Réponse

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

137. Carte 137

     Question

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

     Réponse

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

138. Carte 138

     Question

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

     Réponse

     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. Carte 139

     Question

     When is a chi-square test for homogeneity appropriate?

     Réponse

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

140. Carte 140

     Question

     What is the chi-square test statistic formula?

     Réponse

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

141. Carte 141

     Question

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

     Réponse

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

142. Carte 142

     Question

     When is the sampling distribution of p̂ approximately normal?

     Réponse

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

143. Carte 143

     Question

     What conditions justify a one-proportion z-interval?

     Réponse

     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. Carte 144

     Question

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

     Réponse

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

145. Carte 145

     Question

     What is the one-proportion z-test statistic?

     Réponse

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

146. Carte 146

     Question

     How is a simulation-based p-value estimated?

     Réponse

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

147. Carte 147

     Question

     What does “fail to reject H₀” mean?

     Réponse

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

148. Carte 148

     Question

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

     Réponse

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

149. Carte 149

     Question

     What conditions support the usual model for p̂₁ − p̂₂?

     Réponse

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

150. Carte 150

     Question

     What conditions justify a two-proportion z-interval?

     Réponse

     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. Carte 151

     Question

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

     Réponse

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

152. Carte 152

     Question

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

     Réponse

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

153. Carte 153

     Question

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

     Réponse

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

154. Carte 154

     Question

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

     Réponse

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

155. Carte 155

     Question

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

     Réponse

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

156. Carte 156

     Question

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

     Réponse

     Bias in the estimator.

157. Carte 157

     Question

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

     Réponse

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

158. Carte 158

     Question

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

     Réponse

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

159. Carte 159

     Question

     What two changes widen a confidence interval for a proportion?

     Réponse

     Using a higher confidence level or a smaller sample size.

160. Carte 160

     Question

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

     Réponse

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

161. Carte 161

     Question

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

     Réponse

     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. Carte 162

     Question

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

     Réponse

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

163. Carte 163

     Question

     What is the power of a hypothesis test?

     Réponse

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

164. Carte 164

     Question

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

     Réponse

     At 0, because its mean is p₁ − p₂.

165. Carte 165

     Question

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

     Réponse

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

166. Carte 166

     Question

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

     Réponse

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

167. Carte 167

     Question

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

     Réponse

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

168. Carte 168

     Question

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

     Réponse

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

169. Carte 169

     Question

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

     Réponse

     (r − 1)(c − 1).

170. Carte 170

     Question

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

     Réponse

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

171. Carte 171

     Question

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

     Réponse

     μₓ̄ = μ.

172. Carte 172

     Question

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

     Réponse

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

173. Carte 173

     Question

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

     Réponse

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

174. Carte 174

     Question

     What hypotheses test whether a population mean exceeds 12?

     Réponse

     H₀: μ = 12 versus Hₐ: μ > 12.

175. Carte 175

     Question

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

     Réponse

     Fail to reject H₀ because 0.08 > 0.05.

176. Carte 176

     Question

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

     Réponse

     μ₁ − μ₂.

177. Carte 177

     Question

     Which procedure estimates μ₁ − μ₂ from two independent samples?

     Réponse

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

178. Carte 178

     Question

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

     Réponse

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

179. Carte 179

     Question

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

     Réponse

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

180. Carte 180

     Question

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

     Réponse

     Reject H₀ because 0.004 < 0.01.

181. Carte 181

     Question

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

     Réponse

     σₓ̄ = σ / √n.

182. Carte 182

     Question

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

     Réponse

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

183. Carte 183

     Question

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

     Réponse

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

184. Carte 184

     Question

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

     Réponse

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

185. Carte 185

     Question

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

     Réponse

     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. Carte 186

     Question

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

     Réponse

     √(σ₁²/n₁ + σ₂²/n₂).

187. Carte 187

     Question

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

     Réponse

     √(s₁²/n₁ + s₂²/n₂).

188. Carte 188

     Question

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

     Réponse

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

189. Carte 189

     Question

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

     Réponse

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

190. Carte 190

     Question

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

     Réponse

     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. Carte 191

     Question

     When is the sampling distribution of x̄ approximately normal?

     Réponse

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

192. Carte 192

     Question

     What conditions justify a one-sample t-interval?

     Réponse

     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. Carte 193

     Question

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

     Réponse

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

194. Carte 194

     Question

     What is the one-sample t-test statistic?

     Réponse

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

195. Carte 195

     Question

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

     Réponse

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

196. Carte 196

     Question

     When is x̄₁ − x̄₂ approximately normal?

     Réponse

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

197. Carte 197

     Question

     What conditions justify a two-sample t-interval?

     Réponse

     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. Carte 198

     Question

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

     Réponse

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

199. Carte 199

     Question

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

     Réponse

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

200. Carte 200

     Question

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

     Réponse

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

201. Carte 201

     Question

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

     Réponse

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

202. Carte 202

     Question

     How is a matched-pairs confidence interval analyzed?

     Réponse

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

203. Carte 203

     Question

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

     Réponse

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

204. Carte 204

     Question

     Which observations enter a matched-pairs t-test?

     Réponse

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

205. Carte 205

     Question

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

     Réponse

     Significant at 0.05, but not at 0.01.

206. Carte 206

     Question

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

     Réponse

     At 5.

207. Carte 207

     Question

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

     Réponse

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

208. Carte 208

     Question

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

     Réponse

     Higher confidence or smaller sample sizes.

209. Carte 209

     Question

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

     Réponse

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

210. Carte 210

     Question

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

     Réponse

     There is not convincing evidence that μ₁ exceeds μ₂.

211. Carte 211

     Question

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

     Réponse

     3, because 18/√36 = 3.

212. Carte 212

     Question

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

     Réponse

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

213. Carte 213

     Question

     What conditions justify a one-sample t-test?

     Réponse

     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. Carte 214

     Question

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

     Réponse

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

215. Carte 215

     Question

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

     Réponse

     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. Carte 216

     Question

     What type of variables belong on a scatterplot?

     Réponse

     Two quantitative variables measured on the same observational units.

217. Carte 217

     Question

     What does the correlation coefficient r describe?

     Réponse

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

218. Carte 218

     Question

     What does ŷ = a + bx represent?

     Réponse

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

219. Carte 219

     Question

     What is a residual?

     Réponse

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

220. Carte 220

     Question

     What makes a regression line the least-squares line?

     Réponse

     It minimizes the sum of squared residuals.

221. Carte 221

     Question

     What four features should a scatterplot description address?

     Réponse

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

222. Carte 222

     Question

     What values can r take?

     Réponse

     Any value from −1 to 1, inclusive.

223. Carte 223

     Question

     How is the slope b interpreted in context?

     Réponse

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

224. Carte 224

     Question

     What does a positive residual mean?

     Réponse

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

225. Carte 225

     Question

     What is the least-squares slope formula?

     Réponse

     b = r(sᵧ/sₓ).

226. Carte 226

     Question

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

     Réponse

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

227. Carte 227

     Question

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

     Réponse

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

228. Carte 228

     Question

     How is the intercept a interpreted in context?

     Réponse

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

229. Carte 229

     Question

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

     Réponse

     3, because 21 − 18 = 3.

230. Carte 230

     Question

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

     Réponse

     a = ȳ − bx̄.

231. Carte 231

     Question

     What makes a linear association look strong?

     Réponse

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

232. Carte 232

     Question

     Does r have measurement units?

     Réponse

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

233. Carte 233

     Question

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

     Réponse

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

234. Carte 234

     Question

     What residual-plot pattern supports using a linear model?

     Réponse

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

235. Carte 235

     Question

     What does r² measure in simple linear regression?

     Réponse

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

236. Carte 236

     Question

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

     Réponse

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

237. Carte 237

     Question

     Why should unusual points be checked before interpreting r?

     Réponse

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

238. Carte 238

     Question

     Why is extrapolation risky?

     Réponse

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

239. Carte 239

     Question

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

     Réponse

     Negative, because observed y is less than predicted ŷ.

240. Carte 240

     Question

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

     Réponse

     The point (x̄, ȳ).

241. Carte 241

     Question

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

     Réponse

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

242. Carte 242

     Question

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

     Réponse

     Nothing. Correlation is symmetric.

243. Carte 243

     Question

     What is interpolation?

     Réponse

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

244. Carte 244

     Question

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

     Réponse

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

245. Carte 245

     Question

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

     Réponse

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

246. Carte 246

     Question

     What is an outlier in a scatterplot?

     Réponse

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

247. Carte 247

     Question

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

     Réponse

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

248. Carte 248

     Question

     When can a regression relationship support a causal conclusion?

     Réponse

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

249. Carte 249

     Question

     What units does a residual use?

     Réponse

     The same units as the response variable y.

250. Carte 250

     Question

     What is an influential point in regression?

     Réponse

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

250 cartes

AP Statistics Flashcards: Complete 5-Unit Course Review

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