Use the applet Hypothesis Test for a Mean to investigate the frequency of Type I and Type II errors. For this exercise, use n = 100 and the normal distribution with mean 50 and standard deviation 10.

a. Set the null mean equal to 50 and the alternative to not equal. Run the applet one time. How many times was the null hypothesis rejected at level .05? In this case, the null hypothesis is true. Which type of error occurred each time the true null hypothesis was rejected? What is the probability of rejecting a true null hypothesis at level .05? How does the proportion of times the null hypothesis was rejected compare with this probability?

b. Clear the applet, then set the null mean equal to 47 and keep the alternative at not equal. Run the applet one time. How many times was the null hypothesis not rejected at level .05? In this case, the null hypothesis is false. Which type of error occurred each time the null hypothesis was not rejected? Run the applet several more times without clearing. Based on your results, what can you conclude about the probability of failing to reject the null hypothesis for the given conditions?