393 APPLET Hypothesis Tests for a Proportion 7.4 ACTIVITY You can find the interactive applet for this activity at MyLab Statistics. SECTION 7.4 Hypothesis Testing for Proportions 393 The Hypothesis tests for a proportion applet generates simulations to investigate hypothesis tests for a population proportion. Specify the Sample size, the population proportion (True p), the null value for the proportion (Null p), and the alternative for the test (Alternative). Click Update applet to load the applet with the specified parameters. Click 1 test, 5 tests, or 1000 tests to generate samples of the specified size with a proportion of successes equal to True p. For each sample, a hypothesis test based on the Z-statistic is performed, and the Z-statistic or the P-value (click one) for the test is plotted. Values colored in red represent tests where the null hypothesis is rejected at the specified level of significance. The default level of 0.05 can be changed by adjusting the Level. The table above the graph tracks the cumulative results and shows the Proportion of the hypothesis tests where the null hypothesis was rejected. Press Reset to clear existing results and start a new simulation. EXPLORE Step 1 Specify values for the Sample size, True p, Null p, and Alternative. Step 2 Click 1 test, 5 tests, or 1000 tests to generate hypothesis tests. DRAW CONCLUSIONS 1. Set Sample size = 25 and True p = 0.35. Test the claim that the proportion is equal to 35%. What are the null and alternative hypotheses? Run the simulation for at least 1000 tests. Compare the proportion of null hypothesis rejections for the 0.05 and 0.01 levels. Is this what you would expect? Explain. 2. Set Sample size = 50 and True p = 0.6. Test the claim that the proportion is at least 40%. What are the null and alternative hypotheses? Run the simulation for at least 1000 tests. Compare the proportion of null hypothesis rejections for the 0.05 and 0.01 levels. Perform a hypothesis test for each level. Use the results of the hypothesis tests to explain the results of the simulation. APPLET Test H : p=0.4 vs. H : p<0.4 where the true p=0.46 1 test Sample size 5 tests 1000 tests Update Applet Reset Info Sample size: Null p: True p: Alternative: 30 30 < 0.46 0.4 0 A Z-statistic -4 0 150 4 2 0 -2 6 100 50 P-value Rolls Level 0.05 -1.645 1006 985 11 0.0109 Frequency Z-statistic Critical values Reject null Total Proportion 986 987 988 989 990 991 992 993 994 995 996 997 998 999 1000 1001 1002 1003 1004 1005 1006
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