AP Statistics Flashcards: Complete 5-Unit Course Review
Review all five revised AP Statistics units with 250 original cards on data, study design, probability, inference, and regression.
За това тесте
Review the revised five-unit AP Statistics course with 250 independently written English flashcards. The deck follows the framework effective fall 2026: Exploring One-Variable Data and Collecting Data; Probability, Random Variables, and Probability Distributions; Inference for Categorical Data: Proportions; Inference for Quantitative Data: Means; and Regression Analysis.
What the cards ask you to retrieve
- concept or condition → meaning
- scenario → appropriate method
- representation → interpretation
- result → contextual conclusion
- formula → use
- common error → correction
The order follows Units 1–5, with prerequisite ideas introduced before later inference and regression applications. Every card has the root ap-statistics tag and exactly one unit tag.
What's deliberately left out
This is a compact active-recall review, not a complete course, an official curriculum, or a promise of a particular score. It does not include full free-response questions, timed multiple-choice simulation, calculator-button tutorials, AP Classroom content, copied official examples, or scoring-guideline imitation.
Scope was reviewed against the official AP Statistics course page and the Course and Exam Description effective fall 2026. Check those official sources for current policies, exam details, and later revisions.
Statistical facts and the official course outline are not claimed as original. The CC0 dedication applies to the deck's independently written card wording, organization, and original cover to the extent the contributor can dedicate those elements.
This independent, unofficial deck is not affiliated with, endorsed by, or sponsored by the College Board. AP® and Advanced Placement® are trademarks owned by the College Board. No exam questions, scoring guidelines, curriculum passages, official examples, tables, logos, or trade dress are copied.
Карти в това тесте
Карта 1
Въпрос
What makes a question a statistical investigative question?
Отговор
It anticipates variability in data and can be answered by collecting and analyzing data about a population or process.
Карта 2
Въпрос
What is an observational unit?
Отговор
An individual item or person from which data are collected.
Карта 3
Въпрос
A student's class year is recorded as freshman, sophomore, junior, or senior. What type of variable is this?
Отговор
Categorical. The values name groups rather than measure a numerical amount.
Карта 4
Въпрос
How does a parameter differ from a statistic?
Отговор
A parameter describes a population; a statistic describes a sample.
Карта 5
Въпрос
How is a category's relative frequency calculated?
Отговор
Divide the category count by the total number of observations.
Карта 6
Въпрос
What should the height of a bar represent in a relative-frequency bar chart?
Отговор
The proportion or percentage of observations in that category.
Карта 7
Въпрос
Number of text messages sent in a day: discrete or continuous?
Отговор
Discrete. It is a count with separated possible values.
Карта 8
Въпрос
Which displays preserve individual quantitative data values?
Отговор
Dotplots and stem-and-leaf plots. A histogram groups values into intervals.
Карта 9
Въпрос
What four features should a description of a quantitative distribution address?
Отговор
Shape, center, variability, and unusual features such as gaps or outliers.
Карта 10
Въпрос
Which measure of center is usually better for a strongly right-skewed distribution?
Отговор
The median, because it is resistant to extreme high values.
Карта 11
Въпрос
The values are 3, 5, 5, and 11. What is the mean?
Отговор
- The sum is 24, divided by 4 observations.
Карта 12
Въпрос
The ordered values are 2, 4, 7, 9, 12, and 20. What is the median?
Отговор
8, the average of the two middle values 7 and 9.
Карта 13
Въпрос
How is the interquartile range calculated?
Отговор
IQR = Q3 − Q1. It measures the spread of the middle 50% of the data.
Карта 14
Въпрос
What does a small standard deviation say about a data set?
Отговор
Values typically lie close to the mean.
Карта 15
Въпрос
Which common summaries are resistant to extreme values?
Отговор
The median and IQR are resistant; the mean and standard deviation are not.
Карта 16
Въпрос
In a modified boxplot, where do the whiskers end?
Отговор
At the smallest and largest observed values within the 1.5 × IQR fences; values beyond the fences are plotted separately as potential outliers.
Карта 17
Въпрос
What are the 1.5 × IQR outlier fences?
Отговор
Lower fence = Q1 − 1.5(IQR); upper fence = Q3 + 1.5(IQR). Values beyond them are flagged as potential outliers.
Карта 18
Въпрос
How should two quantitative distributions be compared?
Отговор
Compare shape, center, variability, and unusual features in context, using the same measure or display basis.
Карта 19
Въпрос
What does a z-score of −1.8 mean?
Отговор
The value is 1.8 standard deviations below the mean.
Карта 20
Въпрос
Every observation is converted from meters to centimeters by multiplying by 100. What happens to the mean and standard deviation?
Отговор
Both are multiplied by 100.
Карта 21
Въпрос
What should an investigative question identify so the conclusion has a clear scope?
Отговор
The variable or parameter of interest and the population to which the conclusion may apply.
Карта 22
Въпрос
What is a census?
Отговор
A study that collects data from every member of the population.
Карта 23
Въпрос
What makes a study an experiment?
Отговор
Researchers deliberately assign treatments to experimental units.
Карта 24
Въпрос
How do prospective and retrospective observational studies differ?
Отговор
A prospective study follows units forward and gathers future data; a retrospective study uses data from the past.
Карта 25
Въпрос
What is a confounding variable in an observational study?
Отговор
A variable associated with both the explanatory and response variables that offers an alternative explanation for their relationship.
Карта 26
Въпрос
What study feature supports generalizing results to a population?
Отговор
Random selection from that population.
Карта 27
Въпрос
What makes a study observational?
Отговор
Researchers observe variables without assigning treatments.
Карта 28
Въпрос
What study feature supports a cause-and-effect conclusion?
Отговор
Random assignment of treatments in a well-designed experiment.
Карта 29
Въпрос
What defines a simple random sample of size n?
Отговор
Every possible sample of size n has the same chance of selection.
Карта 30
Въпрос
What changes when sampling is done with replacement?
Отговор
A selected unit returns to the population and can be selected again.
Карта 31
Въпрос
Why can a convenience sample be biased?
Отговор
Easy-to-reach units may differ systematically from the target population.
Карта 32
Въпрос
Why should an experiment compare at least two treatment groups?
Отговор
The comparison provides a baseline for judging whether responses differ by treatment.
Карта 33
Въпрос
A school samples 20 students at random from each grade. Which sampling method is this?
Отговор
Stratified random sampling, with grade as the stratum.
Карта 34
Въпрос
What is the purpose of random assignment?
Отговор
It tends to balance lurking variables across treatment groups, supporting causal inference.
Карта 35
Въпрос
Why can a voluntary-response sample be biased?
Отговор
People with strong opinions are often more likely to participate.
Карта 36
Въпрос
What does replication mean in an experiment?
Отговор
Assigning more than one experimental unit to each treatment so treatment differences can be separated from individual variability.
Карта 37
Въпрос
A city randomly selects 8 apartment buildings and surveys every household in those buildings. Which method is this?
Отговор
Cluster random sampling.
Карта 38
Въпрос
What does direct control do in an experiment?
Отговор
It holds potential extraneous sources of variation constant across experimental units.
Карта 39
Въпрос
What is undercoverage?
Отговор
Some groups in the target population are left out of, or poorly represented in, the sampling frame.
Карта 40
Въпрос
What is the role of a control group?
Отговор
It supplies a comparison condition for evaluating the treatment of interest.
Карта 41
Въпрос
After a random start, a quality inspector checks every 40th item. Which sampling method is this?
Отговор
Systematic random sampling.
Карта 42
Въпрос
Why might an experiment use a placebo?
Отговор
To separate a treatment's effect from responses caused by expecting treatment.
Карта 43
Въпрос
What is nonresponse bias?
Отговор
Selected individuals who do not respond differ in a relevant way from those who do.
Карта 44
Въпрос
What is single blinding designed to reduce?
Отговор
Bias caused when participants or evaluators know which treatment was received, depending on who is blinded.
Карта 45
Въпрос
Why use a randomized block design?
Отговор
To group units that are similar on an important source of variation, then compare treatments within each block.
Карта 46
Въпрос
What defines a matched-pairs design?
Отговор
Two treatments are compared using paired similar units or by giving both treatments to each unit in randomized order.
Карта 47
Въпрос
A survey asks, “Don't you agree the new schedule is unfair?” What problem does this create?
Отговор
Response bias from leading wording.
Карта 48
Въпрос
What usually makes an experiment double-blind?
Отговор
Neither the participants nor the people evaluating responses know treatment assignments while outcomes are measured.
Карта 49
Въпрос
A researcher randomly assigns 80 volunteers to two diets and compares blood-pressure change. What conclusion can random assignment support?
Отговор
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.
Карта 50
Въпрос
A researcher records coffee intake and sleep duration without assigning either. Can the study establish that coffee causes less sleep?
Отговор
No. It is observational, so confounding can provide alternative explanations.
Карта 51
Въпрос
What is the difference between a population and a sample?
Отговор
The population is the full group of interest; a sample is the subset actually observed.
Карта 52
Въпрос
Which graph is appropriate for the distribution of one quantitative variable measured on 600 people?
Отговор
A histogram is appropriate; it groups the many numerical values into intervals.
Карта 53
Въпрос
In a strongly right-skewed distribution, how do the mean and median usually compare?
Отговор
The mean is usually larger because high values pull it to the right.
Карта 54
Въпрос
Every score increases by 7 points. What happens to the mean and standard deviation?
Отговор
The mean increases by 7; the standard deviation stays unchanged.
Карта 55
Въпрос
What does it mean that a score is at the 80th percentile?
Отговор
About 80% of scores are at or below it.
Карта 56
Въпрос
Why should gaps and clusters be mentioned when describing a distribution?
Отговор
They may reveal distinct subgroups, collection effects, or other structure that center and spread alone hide.
Карта 57
Въпрос
What is the minimum ethical safeguard when collecting identifiable human data?
Отговор
Obtain informed consent when required and protect participants' privacy and confidentiality.
Карта 58
Въпрос
Every measurement is multiplied by −2. What happens to the mean and standard deviation?
Отговор
The mean is multiplied by −2; the standard deviation is multiplied by 2.
Карта 59
Въпрос
A study uses random sampling but no assigned treatment. What can it support?
Отговор
Population generalization, but not a cause-and-effect conclusion.
Карта 60
Въпрос
A report calls any unmeasured variable a confounder. What is the correction?
Отговор
A confounder must be related to both the explanatory and response variables and create an alternative explanation.
Карта 61
Въпрос
What does a two-way table summarize?
Отговор
Counts or relative frequencies for combinations of two categorical variables.
Карта 62
Въпрос
What is a joint relative frequency?
Отговор
A cell count divided by the grand total, representing one combination of categories.
Карта 63
Въпрос
What is a marginal relative frequency?
Отговор
A row or column total divided by the grand total.
Карта 64
Въпрос
How is a conditional relative frequency calculated within one row?
Отговор
Divide each cell in that row by the row total.
Карта 65
Въпрос
What pattern suggests association between two categorical variables?
Отговор
The conditional distribution of one variable changes across categories of the other.
Карта 66
Въпрос
Why are segmented bar charts useful for two categorical variables?
Отговор
They place conditional distributions on the same 100% scale, making category patterns easy to compare.
Карта 67
Въпрос
How do an outcome and an event differ?
Отговор
An outcome is one result of a trial; an event is a set of one or more outcomes.
Карта 68
Въпрос
What must a valid probability simulation specify?
Отговор
A chance mechanism whose outcomes match the event probabilities, one trial definition, the statistic recorded, and many repetitions.
Карта 69
Въпрос
What does the law of large numbers predict?
Отговор
As independent trials accumulate, an event's long-run relative frequency tends to approach its probability.
Карта 70
Въпрос
What two requirements must probabilities in a sample space satisfy?
Отговор
Each probability is between 0 and 1, and the probabilities of all nonoverlapping outcomes sum to 1.
Карта 71
Въпрос
What is the complement rule?
Отговор
P(Aᶜ) = 1 − P(A). It is often useful for “at least one” events.
Карта 72
Въпрос
How can you verify that events A and B are mutually exclusive?
Отговор
Their intersection is impossible, so P(A ∩ B) = 0.
Карта 73
Въпрос
What is the formula for P(A | B), when P(B) > 0?
Отговор
P(A | B) = P(A ∩ B) / P(B). The restricted sample space is B.
Карта 74
Въпрос
What is the general multiplication rule for two events?
Отговор
P(A ∩ B) = P(A)P(B | A), or equivalently P(B)P(A | B).
Карта 75
Въпрос
What does it mean for events A and B to be independent?
Отговор
Knowing that one occurred does not change the probability of the other.
Карта 76
Въпрос
What is the general addition rule?
Отговор
P(A ∪ B) = P(A) + P(B) − P(A ∩ B).
Карта 77
Въпрос
Why are two mutually exclusive events with positive probabilities not independent?
Отговор
If one occurs, the other cannot occur, so its conditional probability drops to 0.
Карта 78
Въпрос
What is a random variable?
Отговор
A numerical value determined by the outcome of a random process.
Карта 79
Въпрос
What makes a table a valid discrete probability distribution?
Отговор
It lists every possible value with probabilities from 0 to 1 that sum to 1.
Карта 80
Въпрос
What does a cumulative distribution value F(x) represent?
Отговор
P(X ≤ x), the probability that the random variable is at most x.
Карта 81
Въпрос
How is the expected value of a discrete random variable calculated?
Отговор
Multiply each possible value by its probability and add: E(X) = ΣxP(X = x).
Карта 82
Въпрос
What does the standard deviation of a random variable measure?
Отговор
The typical distance of long-run outcomes from the random variable's mean.
Карта 83
Въпрос
How is the standard deviation of a discrete random variable calculated?
Отговор
σₓ = √[Σ(x − μₓ)²P(X = x)]. The quantity inside the square root is Var(X).
Карта 84
Въпрос
A game has E(X) = −$0.40 per play. What does this mean?
Отговор
Over many plays, the player's average net result approaches a loss of 40 cents per play; it does not predict every play.
Карта 85
Въпрос
What conditions define a binomial random variable?
Отговор
A fixed number of independent trials, two outcomes per trial, constant success probability, and X counts successes.
Карта 86
Въпрос
For X ~ Binomial(n, p), what are the mean and standard deviation?
Отговор
Mean = np; standard deviation = √[np(1 − p)].
Карта 87
Въпрос
For X ~ Binomial(n, p), what is P(X = x)?
Отговор
Choose x success positions, then multiply: C(n, x)pˣ(1 − p)ⁿ⁻ˣ.
Карта 88
Въпрос
How can P(X ≥ 1) be found efficiently for a binomial variable?
Отговор
Use the complement: P(X ≥ 1) = 1 − P(X = 0).
Карта 89
Въпрос
What should one simulated trial represent when estimating P(X ≥ 4) for X ~ Binomial(10, 0.3)?
Отговор
Ten independent success/failure observations with success probability 0.3, followed by recording whether at least four successes occurred.
Карта 90
Въпрос
What features characterize a normal distribution?
Отговор
It is continuous, symmetric, unimodal, and bell-shaped.
Карта 91
Въпрос
Which parameters determine a normal distribution?
Отговор
Its mean μ sets the center, and its standard deviation σ sets the spread.
Карта 92
Въпрос
What is the standard normal distribution?
Отговор
The normal distribution with mean 0 and standard deviation 1.
Карта 93
Въпрос
What is the 68–95–99.7 rule?
Отговор
For an approximately normal distribution, about 68%, 95%, and 99.7% of values lie within 1, 2, and 3 standard deviations of the mean.
Карта 94
Въпрос
What does an area under a normal curve represent?
Отговор
The probability or population proportion within the corresponding interval.
Карта 95
Въпрос
How do you find the value cutting off the lowest 10% of a normal distribution?
Отговор
Find the z-score with cumulative area 0.10, then convert with x = μ + zσ.
Карта 96
Въпрос
A normal variable has μ = 50 and σ = 8. What z-score corresponds to x = 62?
Отговор
1.5, because z = (62 − 50) / 8.
Карта 97
Въпрос
Two exam scores come from different normal distributions. What makes their percentiles comparable?
Отговор
Standardize each score with its own distribution's mean and standard deviation, then compare z-scores or cumulative areas.
Карта 98
Въпрос
What is a sampling distribution of a statistic?
Отговор
The distribution of that statistic over all possible random samples of a fixed size from a population.
Карта 99
Въпрос
How can a sampling distribution be approximated by simulation?
Отговор
Repeatedly take random samples of the same size, calculate the statistic each time, and graph the resulting values.
Карта 100
Въпрос
What is a randomization distribution?
Отговор
A simulated distribution of a statistic produced by repeatedly reallocating responses or labels as specified by a null model.
Карта 101
Въпрос
What does the central limit theorem say about sample means?
Отговор
For random samples, the sampling distribution of the sample mean becomes approximately normal as sample size grows, even when the population is not normal.
Карта 102
Въпрос
How does increasing sample size affect the normal approximation in the central limit theorem?
Отговор
It generally improves the approximation, especially for skewed or irregular populations.
Карта 103
Въпрос
A segmented bar chart shows nearly identical category proportions for every group. What does that suggest?
Отговор
Little or no association between the two categorical variables.
Карта 104
Въпрос
In a survey, 30 of 120 students both bike to school and arrive before 8:00. What is the joint relative frequency?
Отговор
0.25, because 30 / 120 = 0.25.
Карта 105
Въпрос
Why can P(A | B) differ from P(B | A)?
Отговор
They use different restricted sample spaces and usually have different denominators.
Карта 106
Въпрос
If P(A) = 0.4 and P(A | B) = 0.4 with P(B) > 0, what does this indicate?
Отговор
A and B are independent because learning B does not change the probability of A.
Карта 107
Въпрос
If independent events have probabilities 0.6 and 0.5, what is the probability that both occur?
Отговор
0.30, using P(A ∩ B) = P(A)P(B).
Карта 108
Въпрос
A prize is $0 with probability 0.7 and $10 with probability 0.3. What is the expected prize?
Отговор
$3, because 0(0.7) + 10(0.3) = 3.
Карта 109
Въпрос
A machine produces defective items independently with probability 0.02. What distribution models the number of defectives in 50 items?
Отговор
Binomial with n = 50 and p = 0.02.
Карта 110
Въпрос
Heights are approximately normal with μ = 170 cm and σ = 6 cm. About what percent lie from 158 to 182 cm?
Отговор
About 95%, because the interval is μ ± 2σ.
Карта 111
Въпрос
What makes an estimator unbiased?
Отговор
Its sampling distribution is centered at the population parameter it estimates.
Карта 112
Въпрос
For random samples of size n, what is the mean of the sampling distribution of p̂?
Отговор
μₚ̂ = p, where p is the population proportion.
Карта 113
Въпрос
Which procedure estimates one population proportion from a random sample?
Отговор
A one-sample z-interval for a population proportion.
Карта 114
Въпрос
How should a confidence interval for a population proportion be interpreted?
Отговор
We are confident at the stated level that the interval captures the true population proportion, in context.
Карта 115
Въпрос
What hypotheses test whether a population proportion differs from 0.40?
Отговор
H₀: p = 0.40 versus Hₐ: p ≠ 0.40.
Карта 116
Въпрос
What is a p-value?
Отговор
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.
Карта 117
Въпрос
What is the hypothesis-test decision rule using significance level α?
Отговор
Reject H₀ when the p-value ≤ α; otherwise fail to reject H₀.
Карта 118
Въпрос
What is a Type I error?
Отговор
Rejecting H₀ when H₀ is actually true.
Карта 119
Въпрос
What is the mean of p̂₁ − p̂₂ for independent random samples?
Отговор
p₁ − p₂.
Карта 120
Въпрос
Which procedure estimates p₁ − p₂ from two independent samples or randomized groups?
Отговор
A two-sample z-interval for a difference between population proportions.
Карта 121
Въпрос
How should a confidence interval for p₁ − p₂ be interpreted?
Отговор
We are confident at the stated level that the interval captures the true difference p₁ − p₂, in context.
Карта 122
Въпрос
What null hypothesis is standard when testing whether two population proportions differ?
Отговор
H₀: p₁ − p₂ = 0, equivalently p₁ = p₂.
Карта 123
Въпрос
A two-proportion test gives p-value 0.018 at α = 0.05. What decision follows?
Отговор
Reject H₀ because 0.018 < 0.05.
Карта 124
Въпрос
When is a chi-square test for independence appropriate?
Отговор
When one random sample provides two categorical variables and the question asks whether they are associated in one population.
Карта 125
Въпрос
How should a chi-square test p-value be interpreted?
Отговор
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.
250 карти
AP Statistics Flashcards: Complete 5-Unit Course Review
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Карта 126
Въпрос
How do bias and variability differ for an estimator?
Отговор
Bias concerns where the sampling distribution is centered; variability concerns how spread out it is.
Карта 127
Въпрос
What is the standard deviation of p̂ when observations are independent?
Отговор
σₚ̂ = √[p(1 − p) / n].
Карта 128
Въпрос
What is the one-proportion z-interval formula?
Отговор
p̂ ± z*√[p̂(1 − p̂) / n].
Карта 129
Въпрос
What does a 95% confidence level describe?
Отговор
In repeated random sampling with the same method, about 95% of the resulting intervals would capture the true parameter.
Карта 130
Въпрос
Which method tests a claim about one population proportion when its conditions hold?
Отговор
A one-sample z-test for a population proportion.
Карта 131
Въпрос
How does the alternative hypothesis determine a p-value's tail area?
Отговор
A greater-than alternative uses the upper tail, a less-than alternative uses the lower tail, and a not-equal alternative uses both tails.
Карта 132
Въпрос
What wording should follow a rejected null hypothesis?
Отговор
There is convincing statistical evidence for the alternative claim about the population parameter, stated in context.
Карта 133
Въпрос
What is a Type II error?
Отговор
Failing to reject H₀ when Hₐ is actually true.
Карта 134
Въпрос
What is the standard deviation of p̂₁ − p̂₂ for independent samples?
Отговор
√[p₁(1 − p₁)/n₁ + p₂(1 − p₂)/n₂].
Карта 135
Въпрос
What standard error is used in a confidence interval for p₁ − p₂?
Отговор
√[p̂₁(1 − p̂₁)/n₁ + p̂₂(1 − p̂₂)/n₂]; the sample proportions are not pooled.
Карта 136
Въпрос
A confidence interval for p₁ − p₂ contains 0. What does that imply?
Отговор
The interval does not provide convincing evidence of a difference between the population proportions at the corresponding two-sided significance level.
Карта 137
Въпрос
Why is a pooled proportion used in a two-proportion z-test with H₀: p₁ = p₂?
Отговор
The null model assumes both samples share one common population proportion, estimated by combining successes and observations.
Карта 138
Въпрос
How should a p-value for a two-proportion test be stated?
Отговор
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.
Карта 139
Въпрос
When is a chi-square test for homogeneity appropriate?
Отговор
When independent samples or randomized groups are compared on the distribution of one categorical response variable.
Карта 140
Въпрос
What is the chi-square test statistic formula?
Отговор
χ² = Σ[(observed − expected)² / expected], summed over all cells.
Карта 141
Въпрос
What usually happens to an estimator's sampling variability as sample size increases?
Отговор
It decreases; estimates from larger random samples tend to cluster more tightly around the parameter.
Карта 142
Въпрос
When is the sampling distribution of p̂ approximately normal?
Отговор
When the expected counts np and n(1 − p) are both at least 10.
Карта 143
Въпрос
What conditions justify a one-proportion z-interval?
Отговор
Random data; independence, checked with n ≤ 10% of the population when sampling without replacement; and at least 10 observed successes and 10 observed failures.
Карта 144
Въпрос
A 95% confidence interval for p is (0.52, 0.61). What does it say about the claim p = 0.50?
Отговор
The interval excludes 0.50, so the data provide evidence against p = 0.50 in a two-sided test at α = 0.05.
Карта 145
Въпрос
What is the one-proportion z-test statistic?
Отговор
z = (p̂ − p₀) / √[p₀(1 − p₀)/n], using the null proportion p₀ in the standard error.
Карта 146
Въпрос
How is a simulation-based p-value estimated?
Отговор
Find the proportion of simulated null statistics at least as extreme as the observed statistic in the direction of Hₐ.
Карта 147
Въпрос
What does “fail to reject H₀” mean?
Отговор
The data do not provide convincing evidence for Hₐ; it does not prove H₀ true.
Карта 148
Въпрос
With sample size and effect fixed, what often happens when α is lowered?
Отговор
The chance of a Type I error decreases, while the chance of a Type II error increases.
Карта 149
Въпрос
What conditions support the usual model for p̂₁ − p̂₂?
Отговор
Independent random samples or randomized groups, independence within each group, and large enough expected success and failure counts for normal approximation.
Карта 150
Въпрос
What conditions justify a two-proportion z-interval?
Отговор
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.
Карта 151
Въпрос
How does increasing both sample sizes affect a confidence interval for p₁ − p₂?
Отговор
It reduces the standard error and usually narrows the interval when other factors stay the same.
Карта 152
Въпрос
What standard error is used in the two-proportion z-test?
Отговор
√[p̂c(1 − p̂c)(1/n₁ + 1/n₂)], where p̂c is the pooled sample proportion.
Карта 153
Въпрос
A randomized experiment uses volunteers assigned to two treatments. A significant two-proportion test supports what scope?
Отговор
A cause-and-effect conclusion for people similar to the volunteers, not automatic generalization to a broader population.
Карта 154
Въпрос
How is an expected count computed in a two-way table under independence?
Отговор
Expected count = (row total × column total) / grand total.
Карта 155
Въпрос
What conditions justify a chi-square test for a two-way table?
Отговор
Random data; independent observations, including the 10% check when sampling without replacement; and every expected cell count greater than 5.
Карта 156
Въпрос
A sampling distribution is centered away from the true parameter. What problem does this reveal?
Отговор
Bias in the estimator.
Карта 157
Въпрос
If p = 0.30 and n = 100, what does μₚ̂ = 0.30 mean?
Отговор
Across many random samples of 100, the average sample proportion would be 0.30.
Карта 158
Въпрос
For a planned proportion interval with margin of error m, what conservative p-value is used when no prior estimate exists?
Отговор
Use p* = 0.50 in n ≥ (z*/m)²p*(1 − p*) because it gives the largest required sample size.
Карта 159
Въпрос
What two changes widen a confidence interval for a proportion?
Отговор
Using a higher confidence level or a smaller sample size.
Карта 160
Въпрос
Which counts check normality for a one-proportion z-test?
Отговор
Use the null model: np₀ ≥ 10 and n(1 − p₀) ≥ 10.
Карта 161
Въпрос
What is wrong with saying “the p-value is the probability that H₀ is true”?
Отговор
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.
Карта 162
Въпрос
What does “statistically significant at α = 0.01” mean?
Отговор
The p-value is at most 0.01, so H₀ is rejected at that significance level.
Карта 163
Въпрос
What is the power of a hypothesis test?
Отговор
The probability that the test rejects H₀ when a particular alternative is true.
Карта 164
Въпрос
If p₁ = p₂, where is the sampling distribution of p̂₁ − p̂₂ centered?
Отговор
At 0, because its mean is p₁ − p₂.
Карта 165
Въпрос
Why must the order p̂₁ − p̂₂ stay consistent throughout an interval?
Отговор
Changing the order reverses the sign and changes the contextual interpretation of every endpoint.
Карта 166
Въпрос
A 95% interval for p₁ − p₂ is (0.04, 0.15). What conclusion is supported?
Отговор
p₁ is plausibly 0.04 to 0.15 higher than p₂; the interval supports a positive difference.
Карта 167
Въпрос
Which success-failure counts are checked for a two-proportion z-test?
Отговор
Expected counts based on the pooled null proportion: n₁p̂c, n₁(1 − p̂c), n₂p̂c, and n₂(1 − p̂c), each at least 10.
Карта 168
Въпрос
A two-proportion test with Hₐ: p₁ ≠ p₂ fails to reject H₀. What conclusion is valid?
Отговор
There is not convincing evidence that the two population proportions differ.
Карта 169
Въпрос
What are the degrees of freedom for a chi-square test on an r × c table?
Отговор
(r − 1)(c − 1).
Карта 170
Въпрос
A chi-square test for independence has a small p-value. What conclusion is appropriate?
Отговор
There is convincing evidence of an association between the two categorical variables in the population, stated in context.
Карта 171
Въпрос
What is the mean of the sampling distribution of x̄ for random samples from a population with mean μ?
Отговор
μₓ̄ = μ.
Карта 172
Въпрос
Which procedure estimates one population mean when the population standard deviation is unknown?
Отговор
A one-sample t-interval for a population mean.
Карта 173
Въпрос
How should a confidence interval for a population mean be interpreted?
Отговор
We are confident at the stated level that the interval captures the true population mean, in context.
Карта 174
Въпрос
What hypotheses test whether a population mean exceeds 12?
Отговор
H₀: μ = 12 versus Hₐ: μ > 12.
Карта 175
Въпрос
A one-sample t-test gives p-value 0.08 at α = 0.05. What decision follows?
Отговор
Fail to reject H₀ because 0.08 > 0.05.
Карта 176
Въпрос
What is the mean of x̄₁ − x̄₂ for independent random samples?
Отговор
μ₁ − μ₂.
Карта 177
Въпрос
Which procedure estimates μ₁ − μ₂ from two independent samples?
Отговор
A two-sample t-interval for a difference between population means.
Карта 178
Въпрос
How should a confidence interval for μ₁ − μ₂ be interpreted?
Отговор
We are confident at the stated level that the interval captures the true difference μ₁ − μ₂, in context.
Карта 179
Въпрос
What null hypothesis is standard when testing whether two population means differ?
Отговор
H₀: μ₁ − μ₂ = 0, equivalently μ₁ = μ₂.
Карта 180
Въпрос
A two-sample t-test gives p-value 0.004 at α = 0.01. What decision follows?
Отговор
Reject H₀ because 0.004 < 0.01.
Карта 181
Въпрос
What is the standard deviation of x̄ when observations are independent?
Отговор
σₓ̄ = σ / √n.
Карта 182
Въпрос
What is the one-sample t-interval formula for μ?
Отговор
x̄ ± t* × s/√n, with t* based on n − 1 degrees of freedom.
Карта 183
Въпрос
What does a 90% confidence level mean for a mean interval procedure?
Отговор
Across many random samples using the same procedure, about 90% of the intervals would capture the true population mean.
Карта 184
Въпрос
Which procedure tests a claim about one population mean when σ is unknown?
Отговор
A one-sample t-test for a population mean.
Карта 185
Въпрос
How should a one-mean test p-value be interpreted?
Отговор
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.
Карта 186
Въпрос
What is the standard deviation of x̄₁ − x̄₂ for independent samples?
Отговор
√(σ₁²/n₁ + σ₂²/n₂).
Карта 187
Въпрос
What standard error is used in a two-sample t-interval for μ₁ − μ₂?
Отговор
√(s₁²/n₁ + s₂²/n₂).
Карта 188
Въпрос
A confidence interval for μ₁ − μ₂ contains 0. What does that imply?
Отговор
The interval does not provide convincing evidence of a difference between the population means at the corresponding two-sided significance level.
Карта 189
Въпрос
What is the two-sample t-statistic for testing H₀: μ₁ − μ₂ = 0?
Отговор
t = [(x̄₁ − x̄₂) − 0] / √(s₁²/n₁ + s₂²/n₂).
Карта 190
Въпрос
How should a two-mean test p-value be interpreted?
Отговор
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.
Карта 191
Въпрос
When is the sampling distribution of x̄ approximately normal?
Отговор
When the population is approximately normal or the random sample is large enough for the central limit theorem to apply.
Карта 192
Въпрос
What conditions justify a one-sample t-interval?
Отговор
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.
Карта 193
Въпрос
How does increasing sample size affect a confidence interval for μ?
Отговор
It lowers the standard error and usually narrows the interval when confidence level and variability stay comparable.
Карта 194
Въпрос
What is the one-sample t-test statistic?
Отговор
t = (x̄ − μ₀) / (s/√n), with n − 1 degrees of freedom.
Карта 195
Въпрос
A t-test fails to reject H₀. What should the conclusion avoid?
Отговор
Avoid saying H₀ is true; say the data do not provide convincing evidence for Hₐ.
Карта 196
Въпрос
When is x̄₁ − x̄₂ approximately normal?
Отговор
When both populations are approximately normal or both independent random samples are large enough for normal approximations.
Карта 197
Въпрос
What conditions justify a two-sample t-interval?
Отговор
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.
Карта 198
Въпрос
A 95% interval for μ₁ − μ₂ is (−7.2, −1.4). What does it support?
Отговор
μ₁ is plausibly 1.4 to 7.2 units lower than μ₂; the interval supports a negative difference.
Карта 199
Въпрос
What sample-shape condition is checked for a two-sample t-test with small samples?
Отговор
Both sample distributions should be free of strong skewness and outliers unless both populations are known to be approximately normal.
Карта 200
Въпрос
A randomized experiment finds a significant difference in mean response. What can random assignment support?
Отговор
A cause-and-effect conclusion for units like those studied, assuming the experiment was well designed.
Карта 201
Въпрос
A population has μ = 40. What does μₓ̄ = 40 mean for samples of size 25?
Отговор
Across all random samples of 25, the average sample mean is 40.
Карта 202
Въпрос
How is a matched-pairs confidence interval analyzed?
Отговор
Compute one difference for each pair, then use a one-sample t-interval on the population mean difference.
Карта 203
Въпрос
A 95% confidence interval for μ is (18.2, 21.7). What does it say about μ = 22?
Отговор
The interval excludes 22, providing evidence against μ = 22 in a two-sided test at α = 0.05.
Карта 204
Въпрос
Which observations enter a matched-pairs t-test?
Отговор
The within-pair differences, not the two original columns treated as independent samples.
Карта 205
Въпрос
A test reports p-value 0.032. At which common levels is it significant: 0.05 or 0.01?
Отговор
Significant at 0.05, but not at 0.01.
Карта 206
Въпрос
If μ₁ − μ₂ = 5, where is the sampling distribution of x̄₁ − x̄₂ centered?
Отговор
At 5.
Карта 207
Въпрос
Does the standard AP two-sample t procedure require equal population variances?
Отговор
No. It uses separate sample variances in the standard error rather than pooling them.
Карта 208
Въпрос
What two changes usually widen a confidence interval for μ₁ − μ₂?
Отговор
Higher confidence or smaller sample sizes.
Карта 209
Въпрос
Why must the order x̄₁ − x̄₂ match the order μ₁ − μ₂ in the hypotheses?
Отговор
Reversing the order reverses the sign and changes the direction of the claim.
Карта 210
Въпрос
A two-sample test with Hₐ: μ₁ > μ₂ fails to reject H₀. What conclusion is valid?
Отговор
There is not convincing evidence that μ₁ exceeds μ₂.
Карта 211
Въпрос
A population has σ = 18 and random samples have n = 36. What is σₓ̄?
Отговор
3, because 18/√36 = 3.
Карта 212
Въпрос
Why is a t distribution used for inference about a mean when σ is unknown?
Отговор
Replacing σ with the sample standard deviation s adds uncertainty, which the heavier-tailed t distribution accounts for.
Карта 213
Въпрос
What conditions justify a one-sample t-test?
Отговор
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.
Карта 214
Въпрос
What distinguishes a two-sample means procedure from a matched-pairs procedure?
Отговор
Two-sample procedures use independent groups; matched-pairs procedures analyze linked observations through their differences.
Карта 215
Въпрос
How are degrees of freedom handled for a two-sample t procedure?
Отговор
Technology usually uses an approximation based on both sample variances and sizes; a conservative fallback uses the smaller of n₁ − 1 and n₂ − 1.
Карта 216
Въпрос
What type of variables belong on a scatterplot?
Отговор
Two quantitative variables measured on the same observational units.
Карта 217
Въпрос
What does the correlation coefficient r describe?
Отговор
The direction and strength of a linear relationship between two quantitative variables.
Карта 218
Въпрос
What does ŷ = a + bx represent?
Отговор
A linear regression model predicting response y from explanatory variable x.
Карта 219
Въпрос
What is a residual?
Отговор
Observed response minus predicted response: residual = y − ŷ.
Карта 220
Въпрос
What makes a regression line the least-squares line?
Отговор
It minimizes the sum of squared residuals.
Карта 221
Въпрос
What four features should a scatterplot description address?
Отговор
Direction, form, strength, and unusual features such as outliers or clusters.
Карта 222
Въпрос
What values can r take?
Отговор
Any value from −1 to 1, inclusive.
Карта 223
Въпрос
How is the slope b interpreted in context?
Отговор
For each one-unit increase in x, the predicted value of y changes by b units on average.
Карта 224
Въпрос
What does a positive residual mean?
Отговор
The observed response is above the model's predicted response.
Карта 225
Въпрос
What is the least-squares slope formula?
Отговор
b = r(sᵧ/sₓ).
Карта 226
Въпрос
A scatterplot trends downward from left to right. What direction is the association?
Отговор
Negative: larger x-values tend to occur with smaller y-values.
Карта 227
Въпрос
Why can r be near 0 even when two variables are strongly related?
Отговор
Correlation measures only linear association, so a strong curved relationship can have r near 0.
Карта 228
Въпрос
How is the intercept a interpreted in context?
Отговор
It is the predicted response when x = 0, provided x = 0 is meaningful and within the data's scope.
Карта 229
Въпрос
A model predicts 18, and the observed response is 21. What is the residual?
Отговор
3, because 21 − 18 = 3.
Карта 230
Въпрос
How is the least-squares intercept found from the slope?
Отговор
a = ȳ − bx̄.
Карта 231
Въпрос
What makes a linear association look strong?
Отговор
The points lie close to a straight-line pattern, regardless of whether the slope is steep or shallow.
Карта 232
Въпрос
Does r have measurement units?
Отговор
No. Correlation is unitless because it is based on standardized values.
Карта 233
Въпрос
For ŷ = 12 + 2.5x, what is predicted when x = 4?
Отговор
22, because 12 + 2.5(4) = 22.
Карта 234
Въпрос
What residual-plot pattern supports using a linear model?
Отговор
Random scatter around zero with no clear curve, trend, or changing spread.
Карта 235
Въпрос
What does r² measure in simple linear regression?
Отговор
The proportion of variation in the response variable explained by its linear relationship with the explanatory variable.
Карта 236
Въпрос
A scatterplot shows a strong association. Does that establish causation?
Отговор
No. A scatterplot alone cannot rule out confounding or other explanations.
Карта 237
Въпрос
Why should unusual points be checked before interpreting r?
Отговор
Correlation is not resistant; an outlier or influential point can change r substantially.
Карта 238
Въпрос
Why is extrapolation risky?
Отговор
The relationship observed over the data range may not continue beyond that range.
Карта 239
Въпрос
A point lies below the regression line. What sign is its residual?
Отговор
Negative, because observed y is less than predicted ŷ.
Карта 240
Въпрос
Which point always lies on a least-squares regression line with an intercept?
Отговор
The point (x̄, ȳ).
Карта 241
Въпрос
Which variable goes on each axis of a scatterplot used for prediction?
Отговор
The explanatory variable goes on the horizontal x-axis; the response variable goes on the vertical y-axis.
Карта 242
Въпрос
What happens to r if the roles of x and y are swapped?
Отговор
Nothing. Correlation is symmetric.
Карта 243
Въпрос
What is interpolation?
Отговор
Predicting a response for an x-value within the range of observed explanatory values.
Карта 244
Въпрос
A residual plot has a clear U-shape. What is the correction?
Отговор
Do not treat the linear model as adequate; the curved pattern shows systematic structure remains.
Карта 245
Въпрос
A regression has r² = 0.64. What does this mean?
Отговор
About 64% of the variation in the response is explained by its linear relationship with the explanatory variable.
Карта 246
Въпрос
What is an outlier in a scatterplot?
Отговор
A point that falls away from the overall pattern of the other points.
Карта 247
Въпрос
What happens to r when x is converted from centimeters to meters?
Отговор
It stays the same because multiplying by a positive constant does not change standardized linear association.
Карта 248
Въпрос
When can a regression relationship support a causal conclusion?
Отговор
Only when the data come from a well-designed randomized experiment and the conclusion matches its scope.
Карта 249
Въпрос
What units does a residual use?
Отговор
The same units as the response variable y.
Карта 250
Въпрос
What is an influential point in regression?
Отговор
A point whose removal substantially changes the fitted regression line or another key regression result.
250 карти
AP Statistics Flashcards: Complete 5-Unit Course Review
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