Statistical Inference and Hypothesis Testing
Domain 6: Statistical Inference and Hypothesis Testing
This domain tests the logic of testing a claim about a parameter: stating hypotheses, choosing a test statistic and its distribution, comparing with a critical value, and writing the conclusion in the language of the problem.
Core rules
- H₀ states the status quo with equality; H₁ is one-sided (>, <) or two-sided (≠) and fixes the critical region.
- Type I error = rejecting a true H₀, probability α (level of significance); type II error = failing to reject a false H₀, probability β; power = 1 − β.
- Large-sample test for a mean: z = (x̄ − μ₀)/(s/√n) with n ≥ 30; critical values 1.96 (5%, two-tailed), 2.58 (1%, two-tailed), 1.645 (5%, one-tailed).
- Small-sample t test: t = (x̄ − μ₀)/(s/√n), s² = Σ(x − x̄)²/(n − 1), ν = n − 1.
- Two samples: z or t for the difference of means; paired t on the differences d_i with ν = n − 1.
- χ² = Σ(O − E)²/E for goodness of fit (ν = k − 1, or k − 1 − m if m parameters are estimated) and for independence in an r × c table (ν = (r − 1)(c − 1)).
- F = s₁²/s₂² for equality of two variances, with ν₁ = n₁ − 1, ν₂ = n₂ − 1.
- Confidence limits for μ: x̄ ± z·SE or x̄ ± t·SE.
Worked example
A sample of 9 gives x̄ = 52 and s = 6; test H₀: μ = 50 against H₁: μ > 50 at the 5% level. SE = 6/3 = 2, so t = (52 − 50)/2 = 1.00 with ν = 8. The one-tailed 5% critical value is t₀.₀₅(8) = 1.860. Since 1.00 < 1.860 we fail to reject H₀: the excess of 2 units is only one standard error, well within ordinary sampling fluctuation.
Traps
- Using a two-tailed critical value for a one-tailed alternative, or vice versa.
- Dividing by n instead of n − 1 in s².
- Saying H₀ is accepted; the correct phrasing is that we fail to reject it.
- Treating a small p-value as proof that the effect is large or practically important.
- Using the wrong degrees of freedom in χ² when parameters are estimated from the same data.