You can use the Errors and Power applet at www.artofstat.com/webapps.html to investigate the performance of significance tests, to illustrate their long-run behavior when used for many samples. For significance tests, set the null hypothesis as H0: π = 0.50 for a one-sided test withHa:π > 0.50 and sample size 50, and set P(Type I error) = α = 0.05.

The applet shows the null sampling distribution of πˆ and the actual sampling distribution of πˆ for various true values of π. Click on Show Type II error, and it also displays P(Type II error), which is the probability of failing to reject H0 even though it is false.

(a) Set the true value of the proportion to be (i) 0.60,

(ii) 0.70, (iii) 0.80.

What happens to P(Type II error) as

π gets farther from the H0 value?

(b) Set the true value of the proportion to be (i) 0.53,

(ii) 0.52, (iii) 0.51.

What value does P(Type II error) approach as π gets closer to the H0 value?

(c) Fix the true value of the proportion to be 0.60.

Show how P(Type II error) changes for n equal to (i) 50, (ii) 100,

(iii) 200.

(d) Summarize how P(Type II error) depends on π (for fixed n) and on n (for fixed π).