Homework #6 for Stat 610

Due Thursday April 26, in class.

   
1. Let data X_1, X_2, ..., X_n be a random
   sample from an exponential distribution with
   mean theta.  Derive the uniformly most powerful
   level alpha test for testing H_0: theta=1 versus
   H_1: theta > 1.

2.  Consider data X_1, X_2, ..., X_n  that forms
    a random sample froma Normal[0, sigma^2 ] distribution.
    Find a uniformly most powerful level alpha test of 
    H_0: sigma = sigma0 versus H_1: sigma > sigma0.

3. Consider the two sample normal problem with data
   X_1, X_2, ..., X_m ~ iid Normal[ mu1, sigma^2 ] and
   Y_1, Y_2, ..., Y_n ~ iid Normal[ mu2, sigma^2 ].
   Show that generalized likelihood ratio test 
   of H_0: mu1 = mu2 versus H_1: mu1 .neq. mu2 is
   equivalent to the two sample t test.