Exercise 17. Let X1, ..., Xn ~ N(u, 02) be i.i.d. where / is known, but o2 is unknown. Consider the estimate 82 = = E( Xi - 1) 2 i=1 for the variance. a Show that C := (Ai- N) ~ x 2 ( n2 ) . 6 i=1 b) For a E (0, 1), denote the a-quantile of x2(n) by an(a). Using this notation, find numbers an, bn E R such that P(62 bno2) = 2.5%. c) Using these results, develop a test which can be used to test the hypothesis Ho : 62 = of against the alternative H1 : 02 + 0. d) Assume we have n = 100, u - 0 and we have observed o2 = 4.81. Test the hypothesis Ho : 02 - 4 us. H1 : 02 + 4 at significance level 5%