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