Chapter 6.1, Problem 12E

Suppose a certain type of fertilizer has an expected yield per acre of μ₁ with variance σ², whereas the expected yield for a second type of fertilizer is μ₂ with the same variance σ². Let $S_1^2$ and $S_2^2$ denote the sample variances of yields based on sample sizes n₁ and n₂, respectively, of the two fertilizers. Show that the pooled (combined) estimator

$\widehat{{\sigma }^2}\text{ = }\frac{(n_1\text{ - 1})S_1^2\text{ + (}{n}_2\text{ - 1)}{S}_2^2}{n_1\text{ + }{n}_2\text{ - 2}}$

is an unbiased estimator of σ².