Testing of Hypotheses
Testing of Hypotheses asks you to judge a claim about a population using sample evidence. The exam tests null and alternative hypotheses, significance levels, one- and two-tailed tests, z and t statistics, critical regions and decision errors.
Core rules
- H₀ states no change or a specified value; H₁ states the claim being tested. The form of H₁ (≠, >, <) fixes a two-tailed or one-tailed test.
- Type I error: rejecting H₀ when it is true (probability α). Type II error: failing to reject H₀ when it is false (probability β). Power = 1 − β.
- Large-sample test for a mean with known σ: z = (x̄ − μ)/(σ/√n). For small n with σ unknown, use t = (x̄ − μ)/(s/√n) and df = n − 1.
- Reject H₀ when the test statistic falls in the critical region, or when p-value ≤ α.
- Common 5% critical values: z = 1.645 for a one-tailed test and z = 1.96 for a two-tailed test.
Worked example
H₀: μ = 50 against H₁: μ > 50. A sample of n = 64 gives x̄ = 52.5; σ = 8; α = 0.05. SE = 8/√64 = 1. z = (52.5 − 50)/1 = 2.5. Critical value for a right-tailed test at 5% is 1.645. Since 2.5 > 1.645, reject H₀. There is sufficient evidence that μ > 50.
Traps
Using the two-tailed critical value 1.96 for a one-tailed test; writing “accept H₀” instead of “fail to reject H₀”; reversing Type I and Type II errors; using s/√n when σ is known; and comparing the p-value with the test statistic rather than with α.