Applet Exercise In Exercise 10.9(h), you observed that

Chapter 10, Problem 11E

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Problem 11E

Applet Exercise In Exercise 10.9(h), you observed that when the null hypothesis is true, for all sample sizes the proportion of the time H0 is rejected is approximately equal to α the probability of a type I error. If we test H0 : p = .5, Ha : p 7= .5, what happens to the value of βwhen the sample size increases? Set the real value of p to .6 and keep the rest of the settings at their default values (α = .05, n = 15).

a In the scenario to be simulated, what is the only kind of error that can be made?

b Click the button “Clear Summary.” Conduct at least 200 simulations. What proportion of the simulations resulted in type II errors (hover the pointer over the box about “Error” in the lower right portion of the display)? How is the proportion of type II errors related to the proportion of times that H0 is rejected?

c Change n, the number of trials used for each simulated test, to 30 and leave all other settings unchanged. Simulate at least 200 tests. Repeat for n = 50 and n = 100. Click the button “Show Summary.” How do the values of β(.6), the probability of a type II error when p = .6, change as the sample size increases?

d Leave the window with the summary information open and continue with Exercise 10.12.

Reference

Applet Exercise Use the applet Hypothesis Testing (for Proportions) to assess the impact of changing the sample size on the value of α. When you access the applet, the default settings will permit simulations, when the true value of p = .5, of repeated α = .05 level Z -tests for H0 : p = .5 versus Ha : p 7= .5 and n = 15.

a What action qualifies as an “error” in the scenario to be simulated?

b Click the button “Draw Sample” to obtain the results associated with a single sample of size 15. How many successes resulted? What is the value for pˆ? Compute the value of the large-sample test statistic. Does your calculation agree with the value of z given in the table beneath the normal curve? Does the value of z fall in the rejection region? Did the result of this simulation result in an error?

c Click the button “Draw Sample” five more times. How many different values for z did you observe? How many values appeared in the rejection region given by the tails of the normal curve?

d Click the button “Draw Sample” until you obtain a simulated sample that results in rejecting H0. What was the value of pˆ that led to rejection of H0? How many tests did you perform until you first rejected H0? Why did it take so many simulations until you first rejected the null?

e Click the button “Draw 50 Samples” until you have completed 200 or more simulations. Hover the pointer over the shaded box above “Reject” in the bottom bar graph. What proportion of the simulations resulted in rejecting H0?

f Why are the boxes above “Reject” and “Error” of exactly the same height?

g Use the up and down arrows to the right of the “n for sample ” line to change the sample size for each simulation to 20. Click the button “Draw 50 Samples” until you have simulated at least 200 tests. What proportion of the simulations resulted in rejecting H0?

h Repeat the instructions in part (g) for samples of size 30, 40, and 50. Click the button “Show Summary” to see the results of all simulations that you performed thus far. What do you observe about the proportions of times that H0 is rejected using samples of size 15, 20, 30, 40, and 50? Are you surprised by these results? Why?

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