69. Measurement error in a particular situation is normally distributed with mean value m and standard deviation 4. Consider testing H0: m  0 versus Ha:

m  0 based on a sample of n  16 measurements.

a. Verify that the usual test with signi cance level

.05 rejects H0 if either or .

Note: That this test is unbiased follows from the fact that the way to capture the largest area under the z curve above an interval having width 3.92 is to center that interval at 0 (so it extends from

1.96 to 1.96).

b. Consider the test which rejects H0 if either or . What is

a, that is, p(0)?

c. What is the power of the test proposed in

(b) when m.1 and when m.1? (Note that .1 and .1 are very close to the null value, so one would not expect large power for such values.) Is the test unbiased?

x  2.17 x

1.81 x  1.96 x

1.96 x  c x

c

d. Calculate the power of the usual test when m.1 and when m.1. Is the usual test a most powerful test? Hint: Refer to your calculations in

(c). Note: It can be shown that the usual test is most powerful among all unbiased tests.