Question

6. A school system has a high rate of turn-over among new teachers. Specifically, 30% of the teachers that are hired leave within 2 years. The superintendent is concerned about the problem and institutes a program of teacher mentoring that he hopes will improve retention of the teachers. After the first 2 years of the program, he evaluates whether it is working by recording what happened with the 16 teachers who were hired at the start of the program. He finds that 3 of original 16 have left.

a. Complete the relevant hypothesis test, using α = .05.

b. Suppose that the mentoring program actually does improve retention to the point where the true probability of a teacher leaving is actually 10%. What was the power of the principal’s study? What does the number you compute mean in English? Explain the relevance (or lack of relevance) of your power calculation to your conclusion in part ‘a.’

Answer 1

> p=0.1 #true probability of a teacher leaving is actually 10%
> p0=0.3 # teachers that are hired leave within 2 years
> alpha=0.05
n=16
z=(p-p0)/sqrt(p*(1-p)/n)
(Power=pnorm(z-qnorm(1-alpha/2))+pnorm(-z-qnorm(1-alpha/2)))

= 0.7601262

Power is 76%

Power is the probability of rejecting the null hypothesis when, in fact, it is false. Power is the probability of making a correct decision (to reject the null hypothesis) when the null hypothesis is false.

So we had p= 0.76 to reject the null hypothesis.