# Proof that the pooled sample variance is biased when paired comparison is not used when it should be

Given the model for what is meant to be a paired comparison design:

$$y_{ij} = \mu_i + \beta_j + \epsilon_{ij}$$

$$i = 1,2; j= 1, 2, ..., 10$$

Eg. An experiment comparing the different in mean hardness of two hardness testers by sampling from 20 pieces of metal. $$\mu_i$$ is the true mean hardness of tip i, $$\beta_j$$ is the effect on hardness due to the jth specimen, and $$\epsilon_{ij}$$ is the random experimental error.

Assuming that both population variances $$\sigma_1^2$$ and $$\sigma_2^2$$ are equal, how can I prove that:

$$E(S_p^2) = \sigma^2 + \sum_{j=1}^{n}\beta_j^2$$

where

I have only gotten that the pooled variance estimator is unbiased, but clearly that is not the case. The point of this is to show the difference with paired design, which would remove the block effects $$\beta_j$$ as opposed to a typical design of an experiment.

From Design and Analysis of Experiments, Douglas C. Montgomery.