XÌ 1 and S 2 1 are the sample mean and sample variance from a population with mean μ1 and variance Ï 1 2 .

XÌ…₁and S²₁are the sample mean and sample variance from a population with mean μ1 and variance σ₁². Similarly, XÌ…₂and S₂²are the sample mean and sample variance from a second independent population with mean μ₂and variance σ₂². The sample sizes are n₁and n₂, respectively. 

(a) Show that X₁ ˆ’ X₂ is an unbiased estimator of μ₁ ˆ’ μ₂.

(b) Find the standard error of XÌ…₁ ˆ’ XÌ…₂. How could you estimate the standard error?

(c) Suppose that both populations have the same variance; that is, σ²₁ = σ²₂ = σ². Show that

is an unbiased estimator of σ².

(n, 1)S; +(n, 1)S; h + n2 - 2