step by step solution please?

Problem 6 Let 21, ... , Xn be n independent observations on random variables X1,.... Xn, where X; E N(u,0), i = 1,..., n. We know that o = = 3 and u is unknown. Let be the arithmetic mean based on the observed sample 21, ..., Ir. To test the null hypothesis H. : = 10 against alternative hypothesis H:u=11 at significance level 0.05, we use the following decision rule: reject H, if 7 > k, where k is a constant. a) Determine the value of the constant k and the smallest value of n such that, for the alternative u= 11, the power of the test is at least 0.9. (7 p) b) Given that the decision rule above is unchanged, will the power the test decrease, increase or become unchanged if the value u in the alternaitve hypothesis H above will increase? The answer must be motivated. Only answer without motivation gives zero points. (3p) Problem 6 Let 21, ... , Xn be n independent observations on random variables X1,.... Xn, where X; E N(u,0), i = 1,..., n. We know that o = = 3 and u is unknown. Let be the arithmetic mean based on the observed sample 21, ..., Ir. To test the null hypothesis H. : = 10 against alternative hypothesis H:u=11 at significance level 0.05, we use the following decision rule: reject H, if 7 > k, where k is a constant. a) Determine the value of the constant k and the smallest value of n such that, for the alternative u= 11, the power of the test is at least 0.9. (7 p) b) Given that the decision rule above is unchanged, will the power the test decrease, increase or become unchanged if the value u in the alternaitve hypothesis H above will increase? The answer must be motivated. Only answer without motivation gives zero points. (3p)