Applied Statistics and Probability for Engineers

9-4

where B(c; n 1 , p 0 ) is the cumulative binomial distribution. To find the critical value for a given

, we would select the largest csatisfying B(c; n 1 , p 0 ). The type II error calculation is

straightforward. Let p 1 be an alternative value of p, with p 1 p 0. If p p 1 , Xis Bin (n, p 1 ).

Therefore

where B(c; n, p 1 ) is the cumulative binomial distribution.

Test procedures for the other one-sided alternative H 1 : pp 0 and the two-sided alternative

H 0 : p p 0 are constructed in a similar fashion. For H 1 : pp 0 the critical region has the form

xc, where we would choose the smallest value of csatisfying 1 B(c 1, n, p 0 ). For the

two-sided case, the critical region consists of both large and small values. Because cis an integer,

it usually isn’t possible to define the critical region to obtain exactly the desired value of .

To illustrate the procedure, let’s reconsider the situation of Example 9-10, where we wish

to test H 0 : p 0.05 verses H 1 : p0.05. Suppose now that the sample size is n 100 and

we wish to use 0.05. Now from the cumulative binomial distribution with n 50 and

p 0.05, we find that B(0; 100, 0.05) 0.0059, B(1; 100, 0.05) 0.0371, and B(2; 100,

0.05) 0.1183 (Minitab will generate these cumulative binomial probabilities). Since B(1;

100, 0.05) 0.03710.05 and B(2; 100, 0.05) 0.11830.05, we would select c 1.

Therefore the null hypothesis will be rejected if x1. The exact significance level for this

test is 0.0371. To calculate the power of the test, suppose that p 1 0.03. Now

and the power of the test is only 0.1946. This is a fairly small power because p 1 is close to p 0.

0.8054

1

 0.1946

1

 B 1 1; 100, 0.03 2

 1

 B 1 c; n, p 12

1

 B 1 c; n, p 12

P 3 Xc when X is Bin 1 n, p 124

P 1 Type II error when p p 12

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