Collect the sample data and compute the value of the test statistic. In the trash bag case, we have computed the value of the test statistic to be z = 2.20.
Step 5: Calculate the p-value by using the test statistic value. The p-value for testing H0: μ = 50 versus Ha: μ > 50 in the trash bag case is the area under the standard normal curve to the right of the test statistic value z = 2.20. As illustrated in Figure 9.3(b), this area is 1 − .9861 = .0139. The p-value is the probability, computed assuming that H0: μ = 50 is true, of observing a value of the test statistic that is greater than or equal to the value z = 2.20 that we have actually computed from the sample data. The p-value of .0139 says that, if H0: μ = 50 is true, then only 139 in 10,000 of all possible test statistic values are at least as large, or extreme, as the value z = 2.20. That is, if we are to believe that H0 is true, we must believe that we have observed a test statistic value that can be described as a 139 in 10,000 chance. Because it is difficult to believe that we have observed a 139 in 10,000 chance, we intuitively have strong evidence that H0: μ = 50 is false and Ha: μ > 50 is true.
Figure 9.3: Testing H0: μ = 50 versus Ha: μ > 50 by Using Critical Values and the p-Value
Step 6: Reject H0 if the p-value is less than α. Recall that the television network has set α equal to .05. The p-value of .0139 is less than the α of .05. Comparing the two normal curves in Figures 9.3(a) and (b), we see that this implies that the test statistic value z = 2.20 is greater than the critical value z.05 = 1.645. Therefore, we can reject H0 by setting α equal to .05. As another example, suppose that the television network had set α equal to .01. The p-value of .0139 is greater than the α of .01. Comparing the two normal curves in Figures 9.3(b) and (c), we see that this implies that the test statistic value z = 2.20 is less than the critical value z.01 = 2.33. Therefore, we cannot reject H0 by setting α equal to .01. Generalizing these examples, we conclude that the value of the test statistic z will be greater than the critical value zα if and only if the p-value is less than α. That is, we can reject H0 in favor of Ha at level of significance α if and only if the p-value is less than α.
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