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