Recall that for the same sample size the smaller the probability of a Type I error, α, the larger the P(Type II error). Let’s check this for Example 13. There we found P(Type II error) for testing H0: p = 1>3 (astrologers randomly guessing) against Ha: p > 1 > 3 when actually p = 0.50, with n = 116. If we use α = 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 p̂ < 0.435.
b. When p = 0.50, a Type II error has probability 0.08. (By comparison, Example 13 found P(Type II error) = 0.02 when α = 0.05, so we see that P(Type II error) increased when P(Type I error) decreased.)