## Question

###### Problem [26 points total][9 points] Suppose we have a random sample Xi, _, Xn from N(u,02) distribution; where 02 is a known constant. Consider the Z-test of Ho: p = Ho versus the following hypotheses with corresponding critical regions:Hypothesis Ha: u > Lo Ha: H < |o Ha: H # LoCritical region C = {(xX1, Xn): 2 > 2(1 - a)} {(x1, ,Xn): 2 < -2(1 - 0)} {(x1_,xn):Izl > 2(1 - a/2 )}1) ii) iii)where the critical regions are defined in terms of the test statistic Z = (X _ /o)/- J (see s

Problem [26 points total] [9 points] Suppose we have a random sample Xi, _, Xn from N(u,02) distribution; where 02 is a known constant. Consider the Z-test of Ho: p = Ho versus the following hypotheses with corresponding critical regions: Hypothesis Ha: u > Lo Ha: H < |o Ha: H # Lo Critical region C = {(xX1, Xn): 2 > 2(1 - a)} {(x1, ,Xn): 2 < -2(1 - 0)} {(x1_,xn):Izl > 2(1 - a/2 )} 1) ii) iii) where the critical regions are defined in terms of the test statistic Z = (X _ /o)/- J (see slides 1.5). For each case, carefully show that the test has significance level & using probability calculations_ b) [5 points] Using the critical region from the test of the hypotheses in part (a i) , derive a one-sided 90% confidence interval for U. [5 points] Using the critical region from the test of the hypotheses in part (a ii), derive a one ~sided 90% confidence interval for U. [5 points] Using the critical region from the test of the hypotheses in part (a iii) , derive the two-sided 90% confidence interval for U. Is it wider O narrower than the confidence interval you found in problem (6)2 [2 points] Provide the interpretation of a (1 0)100% confidence interval for p_