394 CHAPTER 8 Hypothesis Testing to the left of z = 1.25 is 0.8944 (see Table A-2 row “1.2” and column “.05”), so the area to the right of that test statistic is 1 - 0.8944 = 0.1056. We get P@value = 0.1056. Figure 8-6 shows the test statistic and P-value for this example. (If using technology, the P-value is 0.1059. This P-value is more accurate because it is found using an unrounded test statistic instead of the rounded value of z = 1.25.) Step 7: Because the P-value of 0.1056 (or 0.1059 from technology) is greater than the significance level of a = 0.05, we fail to reject the null hypothesis. Step 8: Because we fail to reject H0: p = 0.5, we do not support the alternative hypothesis of p 7 0.5. Here is the conclusion: There is not sufficient sample evidence to support the claim that most Internet users utilize two-factor authentication to protect their online data. p 5 0.5 or z 5 0 P-value 5 0.1056 p 5 482/926 or z = 1.25 ˆ FIGURE 8-6 P-Value Method p 5 0.5 or z 5 0 a 5 0.05 Test Statistic: z 5 1.25 Critical Value: z 5 1.645 FIGURE 8-7 Critical Value Method Solution: Critical Value Method The critical value method of testing hypotheses is summarized in Figure 8-1 on page 376. When using the critical value method with the claim given in Section 8-1 Example 1, “Most Internet users utilize two-factor authentication to protect their online data,” Steps 1 through 5 are the same as in Steps 1 through 5 for the P-value method, as shown on the previous pages. We continue with Step 6 of the critical value method. Step 6: The test statistic is computed to be z = 1.25, as shown for the preceding P-value method. With the critical value method, we now find the critical values (instead of the P-value). This is a right-tailed test, so the area of the critical region is an area of a = 0.05 in the right tail. Referring to Table A-2 and applying the methods of Section 6-1, we find that the critical value is z = 1.645, which is at the boundary of the critical region, as shown in Figure 8-7. Step 7: Because the test statistic does not fall within the critical region, we fail to reject the null hypothesis. Step 8: Because we fail to reject H0: p = 0.5, we conclude that there is not sufficient sample evidence to support the claim that most Internet users utilize two-factor authentication to protect their online data. (It is very important to use the correct wording for this final statement. See Figure 8-5 for help with wording this final conclusion.) CP
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