9.62 Balancing Type I and Type II errors Recall that for the same sample size, the smaller the probability of a Type I error,

a, the larger the P(Type II error). Let’s check this for Example 12. There we found P(Type II error) for testing H0: p = 1>3 (astrologers randomly guessing) against Ha: p 7 1>3 when actually p = 0.50, with n = 116. If we use a = 0.01, verify that:

a. A Type II error occurs if the sample proportion falls less than 2.326 standard errors above the null hypothesis value, which means pn 6 0.435.

b. When p = 0.50, a Type II error has probability 0.08. (By comparison, Example 12 found P1Type II error2 = 0.02 when a = 0.05, so we see that P(Type II error) increased when P(Type I error)

decreased.)