398 CHAPTER 7 Hypothesis Testing with One Sample Using a Hypothesis Test for the Standard Deviation A company claims that the standard deviation of the lengths of time it takes an incoming telephone call to be transferred to the correct office is less than 1.4 minutes. A random sample of 25 incoming telephone calls has a standard deviation of 1.1 minutes. At a = 0.10, is there enough evidence to support the company’s claim? Assume the population is normally distributed. SOLUTION Because the sample is random and the population is normally distributed, you can use the chi-square test. The claim is “the standard deviation is less than 1.4 minutes.” So, the null and alternative hypotheses are H0: s Ú 1.4 minutes and Ha: s 6 1.4 minutes. (Claim) The test is a left-tailed test, the level of significance is a = 0.10, and the degrees of freedom are d.f. = 25 - 1 = 24. So, using Table 6, the critical value is x 2 0 = 15.659. The rejection region is x 2 6 15.659. The standardized test statistic is x 2 = 1n - 12s2 s 2 Use the chi-square test. = 125 - 1211.122 11.422 Assume s = 1.4. ≈ 14.816. Round to three decimal places. The figure below shows the location of the rejection region and the standardized test statistic x 2. Because x 2 is in the rejection region, you reject the null hypothesis. 20 25 30 35 40 5 10 0 2χ α = 15.659 2χ ≈ 14.816 = 0.10 2χ Interpretation There is enough evidence at the 10% level of significance to support the claim that the standard deviation of the lengths of time it takes an incoming telephone call to be transferred to the correct office is less than 1.4 minutes. TRY IT YOURSELF 5 A police chief claims that the standard deviation of the lengths of response times is less than 3.7 minutes. A random sample of 9 response times has a standard deviation of 3.0 minutes. At a = 0.05, is there enough evidence to support the police chief’s claim? Assume the population is normally distributed. Answer: Page A41 Study Tip Although you are testing a standard deviation in Example 5, the standardized test statistic x 2 requires variance. Remember to square the standard deviation to calculate the variance. EXAMPLE 5
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