8-4 Testing a Claim About a Standard Deviation or Variance 419 YOUR TURN. Do Exercise 7 “Body Temperature.” Step 3: The expression s 6 0.062 g does not contain equality, so it becomes the alternative hypothesis. The null hypothesis is the statement that s = 0.062 g. H0: s = 0.062 g H1: s 6 0.062 g Step 4: The significance level is a = 0.05. Step 5: Because the claim is made about s, we use the x2 (chi-square) distribution. Step 6: The Statdisk display shows the test statistic of x2 = 13.795 (rounded) and it shows that the P-value is 0.0674. Step 7: Because the P-value is greater than the significance level of a = 0.05, we fail to reject H0. INTERPRETATION Step 8: There is not sufficient evidence to support the claim that the listed weights are from a population with a standard deviation that is less than 0.062 g. There isn’t enough evidence to show that the new minting procedure is achieving the goal of reducing the variation of weights of quarters. Critical Value Method Technology typically provides a P-value, so the P-value method is used. If technology is not available, the P-value method of testing hypotheses is a bit challenging because Table A-4 allows us to find only a range of values for the P-value. Instead, we could use the critical value method. Steps 1 through 5 in Example 1 would be the same. In Step 6, the test statistic is calculated by using s = 0.062 g (as assumed in the null hypothesis in Example 1), n = 24, and s = 0.0480164 g, which is the unrounded standard deviation computed from the original list of 24 weights. We get this test statistic: x2 = 1n - 12s2 s 2 = 124 - 12 10.048016422 0.0622 = 13.795 The critical value of x2 = 13.091 is found from Table A-4, and it corresponds to 23 degrees of freedom and an “area to the right” of 0.95 (based on the significance level of 0.05 for a left-tailed test). See Figure 8-11. In Step 7 we fail to reject the null hypothesis because the test statistic of x2 = 13.795 does not fall in the critical region, as shown in Figure 8-11. We conclude that there is not sufficient evidence to support the claim that the listed weights are from a population with a standard deviation that is less than 0.062 g. a 5 0.05 Critical Value: x 2 5 13.091 Test Statistic: x 2 5 13.795 FIGURE 8-11 Testing the Claim That S * 0.062 g
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