Inference for regression slope uses a sample’s slope b to draw conclusions about the true slope β of the population regression line. The test uses t = (b − β₀) ÷ SE_b, and a confidence interval is b ± t* · SE_b, both with df = n − 2. Testing H₀: β = 0 checks for a real linear relationship.
A chi-square test for independence checks whether two categorical variables are related in a two-way table. Find each cell’s expected count, E = (row total × column total) ÷ table total, then add χ² = Σ (O − E)²/E over all cells. Use df = (rows − 1)(columns − 1). A small p-value is evidence of an association.
A chi-square goodness-of-fit test checks whether the counts in one categorical variable fit a claimed distribution. For each category, compare the observed count O with the expected count E, then add up χ² = Σ (O − E)²/E. Use df = number of categories − 1. A large χ² and a small p-value are evidence that the claim is wrong.
A two-sample t-test checks whether two population means differ. You compare the sample means using t = (x̄₁ − x̄₂) ÷ √(s₁²/n₁ + s₂²/n₂). For degrees of freedom, use the smaller of n₁ − 1 and n₂ − 1, or the calculator’s value. A small p-value is evidence that μ₁ and μ₂ are different.
A paired t-test checks whether the mean difference in matched pairs data is zero. You find each pair’s difference d, then run a one-sample t-test on the differences: t = (x̄_d − 0) ÷ (s_d/√n), with n − 1 degrees of freedom, where n is the number of pairs. A small p-value is evidence of a real mean difference.
A one-sample t-test checks a claim about a single population mean μ. You compare the sample mean x̄ with the claimed value μ₀ using t = (x̄ − μ₀) ÷ (s/√n), with n − 1 degrees of freedom. Then you find the p-value from a t-distribution. A small p-value is evidence against the claim.
A two-proportion z-test checks whether two population proportions differ. You compare the sample proportions p̂₁ and p̂₂ using z = (p̂₁ − p̂₂) ÷ √(p̂c(1 − p̂c)(1/n₁ + 1/n₂)), where p̂c is the pooled proportion of successes in both samples. A small p-value is evidence that p₁ and p₂ are different.
Type I and Type II errors are the two ways a hypothesis test can be wrong. A Type I error rejects a true null hypothesis H₀. A Type II error fails to reject a false H₀. The chance of a Type I error is α. Power, 1 − β, is the chance a test catches a false H₀.
A one-proportion z-test checks a claim about a single population proportion p. You compare the sample proportion p̂ with the claimed value p₀ using z = (p̂ − p₀) ÷ √(p₀(1 − p₀)/n). Then you find the p-value from the normal curve. A small p-value is evidence against the claim.
A p-value is the probability of getting a sample result at least as extreme as the one you observed, assuming the null hypothesis is true. A small p-value means your data would be rare if H₀ were true, so it counts as evidence against H₀. If the p-value is at or below the significance level α, you reject H₀.