Page 278 - Probability, Random Variables and Random Processes

270 DECISION THEORY [CHAP. 8

8.4. Consider the binary decision problem of Prob. 8.3. We modify the decision rule such that we
reject H, if x 2 c.

(a) Find the value of c such that the probability of a Type I error a = 0.05.
(b) Find the probability of a Type I1 error /I when p, = 55 with the modified decision rule.

(a) Using the result of part (b) in Prob. 8.3, c is selected such that [see Eq. (8.27)J
a = g(50) = P(x 2 c; p = 50) = 0.05
However, when p = 50, X = N(50; 4), and [see Eq. (8.2811

From Table A (Appendix A), we have 0(1.645) = 0.95. Thus
c - 50
-- - 1.645 and c = 50 + 2(1.645) = 53.29
2
(b) The power function g(p) with the modified decision rule is

Setting p = p, = 55 and using Table A (Appendix A), we obtain

Comparing with the results of Prob. 8.3, we notice that with the change of the decision rule, a is
reduced from 0.1587 to 0.05, but j3 is increased from 0.0668 to 0.1963.

8.5. Redo Prob. 8.4 for the case where the sample size n = 100.
(a) With n = 100, we have

As in part (a) of Prob. 8.4, c is selected so that
a= g(50) = P(8 2 c; p = 50) = 0.05
Since X = N(50; I), we have

Thus c - 50 = 1.645 and c = 51.645
(b) The power function is

Setting p = p, = 55 and using Table A (Appendix A), we obtain
fl = Pn = 1 - g(55) = (D(51.645 - 55) = @(-3.355) x 0.0004