Problem 1. Suppose x1, . . . , x_36 are 36 random samples from N (, 9). We want to test H_0 : = 0 vs = 0.5. If the test rejects H0 if and only of X > 0.2, compute the size and power of this test.

Problem 2. Suppose x1, . . . , x36 are 36 random samples from N (, 9). We want to test H_0 : = 0 vs = 0.5. If the test rejects H0 if and only of X > c. We want to use c such that the size of the test is equal to 0.01. Derive the value of c and then compute the power of this test.

Problem 3. Suppose x1, . . . , x100 are iid samples from Ber(p). We want to test H0 : p = 0.5 vs = p = 0.5. If the test rejects H0 if and only of | X 0.5| > 0.01. Find the size of this test.

Problem 4. Suppose X is a single observation from one of two possible discrete distributions given below: x = 0 x = 1 x = 2 x = 3 x = 4 distribution 0: P (X = x) 0.1 0.3 0.2 0.3 0.1 distribution 1: P (X = x) 0.2 02 0.2 0.2 0.2 If the test rejects H0 if and only if X is even. Find the size and power of this test.

Problem 5. Suppose x1, . . . , x2n are 36 random samples from N (, 1). We want to test H_0 : = 0 vs = 0.5. Student A can only see data x1, . . . , xn and student B can only see data xn+1, . . . , x2n. Both students use the following test: H0 is rejected if and only of X > c and c is chosen such that the size of their test equals . A and B reports their final decision to C, and C decides to reject H0 if and only if both A and B recommend to reject H0. Find the type I and type II errors of C.