Chapter 9.6, Problem 61PB

Gender bias in selecting managers Exercise 9.20 tested the claim that female employees were passed over for management training in favor of their male colleagues. Statewide, the large pool of more than 1000 eligible employees who can be tapped for management training is 40% female and 60% male. Let p be the probability of selecting a female for any given selection. For testing H₀: p = 0.40 against H_a: p < 0.40 based on a random sample of 50 selections, using the 0.05 significance level, verify that:

1. a. A Type II error occurs if the sample proportion falls less than 1.645 standard errors below the null hypothesis value, which means that $\hat{p}>0.286$.
2. b. When p = 0.20, a Type II error has probability 0.06.