Question

Let X1,...,Xn be a random sample from a Normal N(0, σ²). Consider H_o : σ² = 16 vs. H_a: σ² = 4.

a)Use the Neyman Pearson lemma to find the best critical region C*.

b)If n = 10 and the size of the test is fixed as α = 0.10, find the critical region and the power when H_o is false.

Answer 1

MP critical region: (xi,..., ln): f(xi, r2, ..., lnHi) > kf(r1,r2,, n Ho (71, ..., Tn) or, or, (71,..., Tn) where cis such that 7L Xunder Ho Hence MP level α critical region : 7L b. For α-0.10, n-10, xả.90 10-4 10 {(X1 , X2, 〉X; <16 * 4.8652=77.8432} с" ,X10): 10 Power when σ-4-P . X' < 778432|σ2-4-0.9652 (use R code: pchisq(77.8432/4,10))