EXAMPLE 34 Calculating , the probability of a Type II error
ATM network operator Star Systems of San Diego reported that active users of debit cards used them an average of 11 times per month. Suppose we are interested in testing whether people use debit cards on average more than 11 times per month, using level of significance . The hypotheses are
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where represents the population mean debit card usage per month. Suppose we have , and , and from Table 4 (page 500) we have .
Solution
Step 1 We have a right-tailed test, so that
Step 2 Figure 49 shows the normal curve centered at , with labeled.
Step 3 The right-tailed test tells us that equals the area under the normal curve drawn in Step 2 to the left of . This is the shaded area in Figure 49. Area represents probability, so we have
Standardizing with , and :
This is a Case 1 problem from Table 8 of Chapter 6 on page 355.
Thus, . This represents the probability of making a Type II error, that is, of not rejecting the hypothesis that the population mean debit card usage is 11 times per month when in actuality it is 13 times per month.
NOW YOU CAN DO
Exercises 5–16.