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