420 CHAPTER 8 Hypothesis Testing Confidence Interval Method As stated earlier, when testing claims about s or s 2, the P-value method, the critical value method, and the confidence interval method are all equivalent in the sense that they will always lead to the same conclusion. See Example 2. Minting Quarters: Confidence Interval Method EXAMPLE 2 Repeat the hypothesis test in Example 1 by constructing a suitable confidence interval. SOLUTION First, we should be careful to select the correct confidence level. Because the hypothesis test is left-tailed and the significance level is 0.05, we should use a confidence level of 90%, or 0.90. (See Table 8-1 on page 376 for help in selecting the correct confidence level.) Using the methods described in Section 7-3, we can use the sample data listed in Example 1 to construct a 90% confidence interval estimate of s. We use n = 24, s = 0.0480164 g, x2 L = 13.091, and x2 R = 35.172. (The critical values x2 L and x2 R are found in Table A-4. Use the row with df = n - 1 = 23. The 0.90 confidence level corresponds to a = 0.10 and we divide that area of 0.10 equally between the two tails so that the areas to the right of the critical values are 0.95 and 0.05. Refer to Table A-4 and use the columns with areas of 0.95 and 0.05 and use the 23rd row.) B1n - 12s2 x2 R 6 s 6 B1n - 12s2 x2 L B124 - 1210.048016422 35.172 6 s 6 B124 - 1210.048016422 13.091 0.03883 g 6 s 6 0.06365 g With this confidence interval, we cannot support the claim that s 6 0.062 g because the value of 0.062 g is contained within the confidence interval. The confidence interval is telling us that s can be any value between 0.03883 g and 0.06365 g; the confidence interval is not telling us that s is less than 0.062 g. We reach the same conclusion found with the P-value method and the critical value method. FINDING P-VALUES IN TWO-TAILED TESTS Because the x2 distribution is not symmetric, P-values in two-tailed tests are extremely difficult to find. A common procedure is to approximate P-values in two-tailed hypothesis tests by doubling the tail areas, as we have done in the preceding sections of this chapter. Because the x2 distribution becomes more symmetric for larger sample sizes, such approximations get better for larger sample sizes. Doubling the tail areas is the procedure used by Minitab, XLSTAT, Statdisk, and StatCrunch. (SPSS and JMP don’t do hypothesis tests for a claim about the standard deviation or variance.) For two-tailed tests, we always have the option of using the critical value method of testing hypotheses of claims about s or s2.
RkJQdWJsaXNoZXIy NjM5ODQ=