Exercise 4.12 (two-sample t-tests) Consider two independent normal random sam- p
ID: 3364445 • Letter: E
Question
Exercise 4.12 (two-sample t-tests) Consider two independent normal random sam- ples with a common (unknown) variance X1,...,Xn ·N(, 2) and Yi, ,Yn ~ 1. Show that the GLRT for testing Ho : 1 Ha vs. H1 : > 2 at level results in the well-known two-sample one-sided t-test: (a) Derive the generalized likelihood ratio and show that it is a monotone function of the statistic X-Y where S2 -MEI(X-X) usually called the pooled variance estimate, is the unbiased n+m-2 estimator of the common variance 2. (b) Verify that under 1 = 2, T, ~ tnt-m-2. (c) Is it a UMP test? 2. Find the corresponding two-sided two-sample t-test for testing H0 : ,-2 vs. H1 : 12 at level a. Extend it to testing H0 : 1-,- vs. , : 1-2 for a fixed . 3, Exploit the duality between two-sided tests and confidence intervals to derive a 100(1-a)% confidence interval for 1-2.Explanation / Answer
In two sided test we will reject Ho if |T|>Critical point
In confidence interval say (c1, c2) will give same conclusion as the testing problem. That if we reject Ho then the confidence interval will not contain the value of H0.
That is for testing Ho :mu1-mu2=d
If this hypothesis is rejected by test then the confidence interval (c1, c2) will not contain d.
Hence conclusions of these two methods are same.
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