Unbiased Estimator for Mean Response to Treatment


Note: This is essentially Exercise 13.80b in Mathematical Statistics with Applications, 5th. Ed., by Wackerly, Mendenhall, and Scheaffer, and is in the context of ANOVA.

My Work So Far: From the model equations, we have \begin{align*} Y_{ij} &=\mu+\tau_i+\beta_j+\eps_{ij}\\ \overline{Y}_{i\bullet} &=\frac1b\sum_{j=1}^bY_{ij}\\ &=\mu+\tau_i+\overline\beta+\overline\eps_{i\bullet}\\ E\szdp{\overline{Y}_{i\bullet}} &=\mu+\tau_i+E\szdp{\overline\beta}+E\szdp{\overline\eps_{i\bullet}}\\ &=\mu+\tau_i. \end{align*}

My Question: To show that $$\overline{Y}_{i\bullet}$$ is an unbiased estimator for the mean response to treatment $$i,$$ I must show that its expected value is equal to the parameter value. But I don't know what the parameter value is. It feels like this is a simple matter of interpretation: what is the parameter corresponding to the mean response to treatment $$i?$$ Why?

Indeed, $$\bar{Y}_{i.}$$ is an unbiased estimator for the mean response to treatment $$i$$:
$$E\left[ \bar{Y}_{i.} \right]=\frac{1}{b}E\left[ \sum_{j=1}^b{{Y}_{ij}} \right]=\frac{1}{b} \sum_{j=1}^b{E\left[{Y}_{ij} \right]}=\frac{b}{b}E[Y_{ij}]=E[\mu]+E[\tau_i]+E[\beta_j]+E[\epsilon_{ij}]\\=\mu+\tau_i+0+0=\mu+\tau_i$$
That's the mean response to treatment $$i$$ according to the setting of $$\mu$$ and $$\tau_i$$, therefore $$\bar{Y}_{i.}$$ is an unbiased estimator.