Question

Suppose that independent samples ( of sizes n(i) ) are taken from each of k populations and that population (i) is normally distributed with mean mu(i) and variance sigma^2, i = 1, ..., k. That is, all populations are normally distributed with the same variance but with (possibly) different means. Let X(i)bar and s(i)^2, i = 1, ..., k be the respective sample means and variances. Let phi = c(1)mu(1) + c(2)mu(2) + ... + c(k)mu(k), where c(1), c(2), ..., c(k) are given constants.

a) Give the distribution of phi hat = c(1)X(1)bar + c(2)X(2)bar + ... + c(k)X(k)bar.

b) Give the distribution of SSE/sigma^2, where SEE = summation sign from i=1 to k of ( n(i) - 1 )s(i)^2

d) Give the distribution of (phi hat - phi)/sqrt( ( c(1)^2/n(1) ) + ( c(2)^2/n(2) ) + ... + c(k)^2/n(k) )*MSE, where MSE = SSE/( n(1) + n(2) + ... + n(k) - k ).

PS When I used a parenthesis like n(k) I indicated k as the index of n.

Thank you in advance!

Answer 1

1) The samples that are drawn from normally distributed populations follow normal distribution and the means of the samples also follows normal distribution.

Hence the given phi hat = c(1)X(1)bar + c(2)X(2)bar + ... + c(k)X(k)bar. follows a normal distribution.

2) SSE/sigma^2

Here the above ratio follows F-Distribution because SSE is calculated from the square of the standard deviations from normal samples and sigma^2 is the norma population variance. The ratio of the square of the two independent normal random variables follow F-Distribution. Hence SSE/sigma^2 follows F-Distribution.

3) The distribution of (phi hat - phi)/sqrt( ( c(1)^2/n(1) ) + ( c(2)^2/n(2) ) + ... + c(k)^2/n(k) )*MSE, is Chi square distribution.

(phi hat - phi)/sqrt( ( c(1)^2/n(1) ) + ( c(2)^2/n(2) ) + ... + c(k)^2/n(k) ) is a constant term and MSE is the squared term which is calculated from normally distributed samples.