Suppose X1 X2 Xn and Y1 Y2 Ym are two independent random sa

Suppose X_1, X_2, ....., X_n and Y_1, Y_2,.... Y_m are two independent random samples from the respective normal distribution N(mu_1, sigma_1^2) and N(mu_1, sigma_1^2) and N(mu_2, sigma_2^2) where the four parameters are unknown. Let S_1^2 and S_2^2 be the respective sample variances. Denote F = S_2^2 /sigma_2^2 / S_1^2 / sigma_1^2 What is the distribution of random variable F = S_2^2 / sigma_2^2 / S_1^2 /. sigma_1^2? Using the random variable F in part (a) as a pivot random variable, find a (1 - alpha)100% confidence interval for sigma_1^2 / sigma_2^2.

Solution

(a):Given X~N(₁,²₁) and Y~N(₂,²₂), under the null hypothesis as H₀:²₁not = to ²_2, F=S²₂/S²₁~F(m-1,n-1). (b): (1-alpha)100% confidence interval for ²₁/²₂ is [-F_alpha(S²₂/S²₁),F_alpha(S²₂/S²₁)]