Suppose that independent samples (of sizes ni) are taken
Chapter 7, Problem 39E(choose chapter or problem)
Suppose that independent samples (of sizes \(\mathrm{n}_{\mathrm{i}}\)) are taken from each of k populations and that
population i is normally distributed with mean \(\mu \text { i }\) and variance
\(\sigma^{2}, \mathrm{i}=1,2, \ldots, \mathrm{k}\).That is,
all populations are normally distributed with the same variance but with (possibly) different means. Let \(\bar{X}_{i}\) and \(S_{i}^{2}, \mathrm{i}=1,2, \ldots, \mathrm{k}\) be the respective sample means and variances.
Let \(\theta=c_{1} \mu_{1}+c_{2} \mu_{2}+\cdots+c_{k} \mu_{k}\) , where
\(\mathrm{C}_{1}, \mathrm{C}_{2}, \ldots, \mathrm{C}_{\mathrm{k}}\) are given constants.
Give the distribution of \(\widehat{\theta}=c_{1} \bar{X}_{1}+c_{2} \bar{X}_{2}+\ldots+c_{k} \bar{X}_{k}\). Provide reasons for any claims that you make.Give the distribution of
\(\frac{S S E}{\sigma^{2}}\), Where \(\text { SSE }=\sum_{i=1}^{k}\left(n_{i}-1\right) S_{i}^{2}\)
Provide reasons for any claims that you make.
Give the distribution of
\(\frac{\hat{\theta}-\theta}{\sqrt{\left(\frac{c_{1}^{2}}{n_{1}}+\frac{c_{2}^{2}}{n_{2}}+\cdots+\frac{c_{k}^{2}}{n_{k}}\right) \operatorname{MSE}}}\)
Where
\(M S E=\frac{S S E}{n_{1}+n_{2}+\ldots+n_{k}-k}\)
Provide reasons for any claims that you make.
Equation Transcription:
Text Transcription:
n_i
\mu i
\sigma^2,i=1,2, \ldots, k
\bar X_i
S_i^2, i=1,2, \ldots,k
\theta=c_1 \mu_1+c_2 \mu_2+\cdots+c_k \mu_k
C_1,C_2, \ldots,C_k
\widehat \theta=c_1 \bar X_1+c_2 \bar X_2+\ldots+c_k \bar X_k
S S E \sigma^2
SSE =\sum_i=1^k(n_i-1) S_i^2
\hat \theta-\theta \sqrt (c_1^2n_1+c_2^2 n_2+\cdots+\c_k^2 n_k \MSE
M S E=S S En_1+n_2+\ldots+n_k-k
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