Applet Exercise Refer to Exercise 10.9. Click the button

Chapter 10, Problem 10E

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Problem 10E

Applet Exercise Refer to Exercise 10.9. Click the button “Clear Summary” to delete the results of any previous simulations. Change the sample size for each simulation to n = 30 and leave the null and alternative hypotheses at their default settings H0 : p = .5, Ha : p 7= .5.

a Leave the true value of p at its default setting p = .5.With this scenario, what is an error? Simulate at least 200 tests. What proportion of the tests resulted in rejecting H0? What do you notice about the heights of the boxes above “Reject” and “Error” in the bottom right-hand graph? Why?

b Leave all settings unchanged except change the true value of p to .6.With this modification, what is an error? Simulate at least 200 tests. What proportion of the tests resulted in rejecting H0? What do you notice about the heights of the boxes above “Reject” and “Error” in the bottom right-hand graph? Why?

c Leave all settings from part (b) unchanged except change the true value of p to .7. Simulate at least 200 tests. Repeat, setting the true value of p to .8. Click the button “Show Summary.” As the true value of p moves further from .5 and closer to 1, what do you observe about the proportion of simulations that lead to rejection of H0? What would you expect to observe if a set of simulations was conducted when the true value of p is .9?

d What would you expect to observe if simulations were repeated when the real value of p is .4, .3, and .2? Try it.

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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