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