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Translate a research claim into hypotheses about a population parameter, put equality in the null hypothesis, choose the alternative direction before seeing the data, and keep sample statistics out of the hypotheses.
Connect decisions in a significance test to their possible consequences: a Type I error rejects a true null, a Type II error fails to reject a false null, and power is the probability of detecting a specified real effect.
Separate three linked but different ideas: the test statistic measures how far the sample result is from the null model, the p-value measures tail evidence under that model, and alpha supplies the decision threshold.
Test a claim about one population proportion using the null value in the standard error, match the p-value tail to the alternative hypothesis, and state the conclusion about the population parameter.
Use a two proportion z test when two independent groups produce binary outcomes and the question asks whether their population proportions differ or differ in a specified direction. The equality null pools the two samples because the null model treats them as sharing one population proportion.
Use a one sample t test to test a claim about one population mean when the population standard deviation is unknown. The procedure standardizes the distance between the sample mean and the null mean with s divided by the square root of n and evaluates that distance on a t distribution.
Choosing between a two-sample t test and a paired t test is a design decision before it is a calculation. Independent groups require a two-sample procedure; matched people, repeated measurements, or deliberately paired units require a one-sample t test on the within-pair differences.
The chi square goodness of fit test compares one categorical variable’s observed counts with a fully specified distribution of expected proportions. College Board removed this procedure from the revised AP Statistics course beginning in 2026–27, so this page is retained for older released material, enrichment, and college-statistics continuity.
Chi-square tests of homogeneity and independence use the same expected-count formula and chi-square statistic but begin from different data-collection stories. Homogeneity compares categorical distributions across populations or treatments; independence asks whether two categorical variables are associated within one population.
Chi-square calculator with expected counts, df, critical values, p-value logic, worked cases, MCQs, FRQs, and a 20-method statistics engine.
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