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d Leave the window with the summary information open and continue with Exercise 10.12.
Exercises 503 10.12 Applet Exercise Refer to Exercise 10.11. Change α to .1 but keep H 0 : p = . 5 , H a : p ?= . 5 and the true value of p = . 6. Simulate at least 200 tests when n = 15. Repeat for n = 30 , 50, and 100. Click on the button “Show Summary.” You will now have two summary tables (it might be necessary to drag the last table from on top of the first). Compare the error rates when tests are simulated using 15, 30, 50, and 100 trials. a Which of the two tests α = . 05 or α = . 10 gives the smaller simulated values for β , using samples of size 15? b Which gives the smaller simulated values for β for each of the other sample sizes? 10.13 Applet Exercise If you were to repeat the instructions of Exercise 10.10, using n = 100 instead of n = 30, what would you expect to be similar? What would you expect to be different? 10.14 Applet Exercise Refer to Exercise 10.9.Setup the appletto test H 0 : p = . 1versus H a : p < . 1 by clicking the radio button “Lower” in the line labeled “Tail” and adjusting the hypothesized value to .1. Set the true value of p = . 1, n = 5, and α = . 20. a Click the button “Draw Sample” until you obtain a sample with zero successes. What is the value of z ? What is the smallest possible value for z ? Is it possible that you will get a sample so that the value of z falls in the rejection region? What does this imply about the probability that the “large sample” test procedure will reject the null hypothesis? Does this result invalidate the use of large sample tests for a proportion? b Will the test from part (a) reject the true null approximately 20% of the time if we use n = 10? Try it by simulating at least 100 tests. What proportion of the simulations result in rejection of the null hypothesis? c Look through the values of ˆ p in the table under the normal curve and identify the value of ˆ p for which the null is rejected. Use the tables in the appendix to compute the probability of observing this value when n = 10 and p = . 1. Is this value close to .2? d Is n = 100 large enough so that the simulated proportion of rejects is close to .2? Simulate at least 100 tests and give your answer based on the simulation. 10.15 Applet Exercise Refer to Exercise 10.10. Click the button “Clear Summary” to delete the results of any previous simulations. Change the sample size for each simulation to n = 30 and set up the applet to simulate testing H 0 : p = . 4 versus H a : p > . 4 at the .05 level of significance. a Click the button “Clear Summary” to erase the results or any previous simulations. Set the real value of p to .4 and implement at least 200 simulations. What is the percentage simulated tests that result in rejecting the null hypothesis? Does the test work as you expected? b Leave all settings as they were in part (a) but change the real value of p to .5. Simulate at least 200 tests. Repeat when the real value of p is .6 and .7. Click the button “Show Summary.” What do you observe about the rejection rate as the true value of p gets further from .4 and closer to 1? Does the pattern that you observe match your impression of how a good test should perform?
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