Page 276 - Schaum's Outlines - Probability, Random Variables And Random Processes

CHAP. 81 DECISION THEORY

Now both hypotheses are simple. We make a Type I1 er.ror if X > 5 when in fact p = 0.02. Hence, by
Eq. (2.37),

Again using the Poisson approximation with L = np = 200(0.02) = 4, we obtain

.
8.3. Let (XI, . . , X,) be a random sample of a normal r.v. X with mean p and variance 100. Let
H,: p= 50
HI: p=p, (>SO)
and sample size n = 25. As a decision procedure, we use the rule to reject H, if ;F 2 52, where E is
the value of the sample mean X defined by Eq. (7.27).
(a) Find the probability of rejecting H,: p = 50 as a function of p (> 50).
(b) Find the probability of a Type I error a.
(c) Find the probability of a Type I1 error /I (i) when pl = 53 and (ii) when p, = 55.
(a) Since the test calls for the rejection of H,: p = 50 when 2 2 52, the probability of rejecting H, is given
by

Now, by Eqs. (4.1 12) and (7.27), we have

Thus, .X is N(p; 4), and using Eq. (2.55), we obtain

The function g(p) is known as the power function of the test, and the value of g(p) at p = p,, g(p,), is
called the power at p,.
(b) Note that the power at p = 50, g(50), is the probability of rejecting H,: p = 50 when H, is true-that
is, a Type I error. Thus, using Table A (Appendix A), we obtain

(c) Note that the power at p = p,, g(pl), is the probability of rejecting H,: p = 50 when p = p,. Thus,
1 - g(p,) is the probability of accepting Ho when p = p,--that is, the probability of a Type I1 error jl.
(i) Setting p = p, = 53 in Eq. (8.28) and using Table A (Appendix A), we obtain

(ii) Similarly, for p = p1 = 55 we obtain

Notice that clearly, the probability of a Type I1 error depends on the value of p,.