Test μ ₁ verses μ ₂, Variances Unknown

Abstract

We have two populations: Pop I and Pop II. Pop I is normally distributed with unknown mean μ ₁ and unknown variance σ ²₁ . Pop II is also normally distributed with unknown mean μ ₂ and unknown variance σ ²₂ . We wish to do the following statistical test

$$\mathop H\nolimits_0 :\mathop \mu \nolimits_1 - \mathop \mu \nolimits_1 = 0$$

(17.1)

verses

$$\mathop H\nolimits_0 :\mathop \mu \nolimits_1 - \mathop \mu \nolimits_1 \ne 0$$

(17.2)

We collect a random sample of size n ₁ from Pop I and let ${{\bar{x}}_{1}}$ be the mean for this data and s ²₁ is the sample variance. We also gather a random sample of size n ₂ from Pop II and ${{\bar{x}}_{2}}$ is the mean for the second sample with s ²₂ the variance. We assume these two random samples are independent.