Now suppose we wish to make it easier to reject the null hypothesis, so we instead set a: = 0.05. Note that qnorm(0.95) = 1.64. 9. (2) What is the probability of making a type-II error now, given that p: = 15? (Hint: Repeat the same steps as in 3-8.) 10. (1) What is the probability of making a type-I error now? 11. (1) Examine your answers to 7-10. What is the relationship between the probability of a type-I error and the probability of a type-II error? Explain why this is, again, as if you're talking to someone who has never taken a probability or statistics class. We've talked a good deal about type-I errors, and a little bit about type-II errors. In this problem, you will suss out the relationship between the two. Suppose X1,. ..,X,, ~ iid N01, 1), and we would like to test H0 : y = 1 0 against the one-sided alternative, H1 : 'u > 1 0. Let X\" be the sample mean. 1. (1) In one sentence or less, dene a type-II error in this setting, in terms of H0 and H1. 2. (1) What is the appropriate test statistic, T, for this problem? What is its sampling distribution under the null hypothesis? For parts 38, suppose we choose a : 0.025. 3. (1) For which values of T do we fail to reject H0? 4. (1) Given your answer to 3, for which values of 22,, do we fail to reject H0? 5. (1) Now suppose the alternative hypothesis H1 is true, and in particular, ,u = IHS. Construct a new statistic using X\" that is N(O,1) under this scenario, and call this T. 6. (1) Given your answer to 4, for which values of T do we fail to reject H0? 7. (1) Again assuming ,u = 15, what is the probability of making a typeII error? Express your answer using pnorm. (Hint: Use T.) 8. (1) What is the probability of a typeI error