Two-sample tests One of the basic problems in elementary statistics is testing for equality of two means. If yj, j = 0,1, are the sample means, the sample sizes are mj, j = 0, 1, and the sample...

Two-sample tests One of the basic problems in elementary statistics is testing for equality of two means. If yj, j = 0,1, are the sample means, the sample sizes are mj, j = 0, 1, and the sample standard deviations are SDj, j = 0, 1, then under the assumption that sample j is NID(μ_j, σ₂), the statistic

With
  is used to test μ₀
= μ₁
against a general alternative. Under normality and the assumptions of equal variance in each population, the null distribution is

 For simplicity assume m₀
= m₁
= m, although the results do not depend on the equal sample sizes. Define a predictor X with values x_i
= 0 for i = 1, . . . , m and x_i
= 1 for i = m+1, . . . , 2m. Combine the response y_i
into a vector of length 2m, the first m observations corresponding to population 0 and the remaining to population 1. In this problem we will fit the simple linear regression model (2.1) for this X and Y, and show that it is equivalent to the two-sample problem.