Please review the question below.

Q6 Q6 18 Points Problem 6. [18 pts] Answer the following questions by your understanding of the facts. Part (I) (a) [3 pts] Explain why probability of type I error will not be affected by the choice of test statistic. (b) [3 pts] Explain why probability of type II error will be affected by the choice of test statistic. (c) [3 pts] Explain why the power of a test will increase if the probability of type I error increases. Part (1I): Recall that duality of confidence interval and hypothesis testing means the equivalence of the two statements: (1) We fail to reject Ho : 0 = 00 in a two-sided test; (2) The confidence interval for the parameter e contains the pre-hypothesized value do. (a) [3 pts] Explain why such a duality holds for one-sample z test and one-sample t test. (b) [3 pts] Explain why such a duality does not hold for one-proportion z test. (c) [3 pts] Does the duality apply to x2 test for population variance? Justify your