t-test assumptions; two means small sample

Assumptions for the t-test

1. The population(s) from which the sample is drawn is (are) normally distributed.
2. The sample is small (n < 30) and is a random sample; observations are independent.
3. The population standard deviation σ is unknown (estimated by the sample standard deviation s).
4. For two-sample t-test: the two samples are independent and the population variances are equal (σ₁² = σ₂²).
5. The data are measured on an interval or ratio scale.

Test of significance for the difference of two means (small samples)

Let samples of sizes n₁ and n₂ (each small) come from normal populations with means μ₁, μ₂ and equal unknown variance.

1. Hypotheses: H₀: μ₁ = μ₂. H₁: μ₁ ≠ μ₂ (two-tailed) or μ₁ > μ₂ / μ₁ < μ₂ (one-tailed).
2. Level of significance: α (usually 0.05).
3. Test statistic: t = (x̄₁ − x̄₂) / [S_p √(1/n₁ + 1/n₂)] where the pooled standard deviation is S_p² = [(n₁ − 1)s₁² + (n₂ − 1)s₂²] / (n₁ + n₂ − 2)
4. Degrees of freedom: ν = n₁ + n₂ − 2.
5. Critical value: t_α/2(ν) for a two-tailed test (t_α(ν) for a one-tailed test) from the t-table.
6. Decision: reject H₀ if |t| > critical value; otherwise do not reject H₀.
7. Conclusion: state whether the two population means differ significantly.

If the variances are not equal, Welch's test with t = (x̄₁ − x̄₂)/√(s₁²/n₁ + s₂²/n₂) and adjusted degrees of freedom is used. For paired (dependent) samples, use t = d̄/(s_d/√n) with n − 1 degrees of freedom.