Hypothesis definition; two means large sample

Hypothesis

A statistical hypothesis is an assumption or statement about a population parameter (or distribution) which is tested on the basis of sample evidence. There are two kinds:

• Null hypothesis H₀: the statement of no difference or no effect (e.g. μ₁ = μ₂); it is the hypothesis tested.
• Alternative hypothesis H₁: the statement accepted if H₀ is rejected (e.g. μ₁ ≠ μ₂, μ₁ > μ₂ or μ₁ < μ₂).

Testing the difference between two population means (large samples)

Independent random samples of sizes n₁, n₂ (both ≥ 30) give means x̄₁, x̄₂ and standard deviations s₁, s₂.

1. State the hypotheses: H₀: μ₁ = μ₂; H₁: μ₁ ≠ μ₂ (two-tailed) or μ₁ > μ₂ / μ₁ < μ₂ (one-tailed).
2. Choose the level of significance α (0.05 or 0.01).
3. Select the test statistic: by the central limit theorem the statistic is standard normal: Z = (x̄₁ − x̄₂) / √(s₁²/n₁ + s₂²/n₂) (σ₁², σ₂² replace s₁², s₂² when known.)
4. Find the critical region:
   • Two-tailed: |Z| > 1.96 (α = 0.05) or |Z| > 2.576 (α = 0.01)
   • Right-tailed: Z > 1.645 (0.05) or 2.326 (0.01); left-tailed: Z < −1.645 or −2.326.
5. Compute Z from the sample data.
6. Decision: reject H₀ if the calculated Z falls in the critical region; otherwise do not reject.
7. Conclusion: interpret in the context of the problem.

Equivalent: the (1 − α) confidence interval (x̄₁ − x̄₂) ± Z_α/2·SE can be used; H₀ is rejected if 0 lies outside the interval.