Refer to the model for the randomized block design with

Chapter 13, Problem 90SE

(choose chapter or problem)

Refer to the model for the randomized block design with random block effect given in Exercise
a Give the expected value and variance of \(Y_{i j}\).
b Let \(\bar{Y}_{i *}\) denote the average of all of the responses to treatment . Use the model for the randomized block design to derive \(E\left(\bar{Y}_{i *}\right)\) and \(V\left(\bar{Y}_{i *}\right)\) Is \(\overline{Y_{i}}\) an unbiased estimator for the mean response to treatment  ? Why or why not? Notice that \(\left(\bar{Y}_{i *}\right)\) depends on  and both \(\sigma_{\beta}^{2}\) and \(\sigma_{\varepsilon}^{2}\).
c  Consider \(\bar{Y}_{i *}-\bar{Y}_{i^{\prime}}\), for \(i \neq i\) Show that \(E\left(Y_{i *}-Y_{i^{\prime} *}\right)=\boldsymbol{\tau}_{i}-\boldsymbol{\tau}_{i^{\prime}}\) This result implies that
, is an unbiased estimator of the difference in the effects of treatments  and
d Derive \(V\left(\bar{Y}_{i *}-\bar{Y}_{i *}\right)\) Notice that \(V\left(\bar{Y}_{i *}-\bar{Y}_{i *}\right)\) depends only on  and \(\sigma_{\varepsilon}^{2}\).

Equation transcription:

Text transcription:

Y{i j}

bar{Y}{i *}

E(\bar{Y}{i *})

V(bar{Y}{i *})

overline{Y{i}}

(\bar{Y}{i *})

sigma{beta}^{2}

sigma{varepsilon}^{2}

bar{Y}{i *}-bar{Y}{i^{\prime}}

i neq i

E(Y{i *}-Y{i^{prime} *})=boldsymbol{tau}{i}-boldsymbol{tau}{i^{\prime}}

V(bar{Y}{i *}-\bar{Y}{i *})