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