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