Let X1, X2 be iid uniform(0, 0 + 1). For testing Ho : 0 = 0 versus H1 : 0 > 0, we have

two competing tests:

$1 (X1) :Reject Ho if X1 > .95,

$2(X1, X2) :Reject Ho if X, + X2 > C.

(a) Find the value of C so that 02 has the same size as $1.

(b) Calculate the power function of each test. Draw a well-labeled graph of each power

function.

c) Prove or disprove: $2 is a more powerful test than $1.

d) Show how to get a test that has the same size but is more powerful than $2.