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