SECTION 7.5 Hypothesis Testing for Variance and Standard Deviation 397 Using a Hypothesis Test for the Population Variance A dairy processing company claims that the variance of the amount of fat in the whole milk processed by the company is no more than 0.25. You suspect this is wrong and find that a random sample of 41 milk containers has a variance of 0.27. At a = 0.05, is there enough evidence to reject 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 variance is no more than 0.25.” So, the null and alternative hypotheses are H0: s 2 … 0.25 (Claim) and Ha: s 2 7 0.25. The test is a right-tailed test, the level of significance is a = 0.05, and the degrees of freedom are d.f. = 41 - 1 = 40. So, using Table 6, the critical value is x 2 0 = 55.758. The rejection region is x 2 7 55.758. The standardized test statistic is x 2 = 1n - 12s2 s 2 Use the chi-square test. = 141 - 1210.272 0.25 Assume s 2 = 0.25. = 43.2. The figure at the left shows the location of the rejection region and the standardized test statistic x 2. Because x 2 is not in the rejection region, you fail to reject the null hypothesis. You can check your answer using technology, as shown below. Note that the test statistic, 43.2, is the same as what you found above. STATCRUNCH One sample variance summary hypothesis test: s2 : Variance of population H0 : s2 = 0.25 HA : s2 7 0.25 Hypothesis test results: Variance Sample Var. DF Chi-square Stat P-value s2 0.27 40 43.2 0.3362 Interpretation There is not enough evidence at the 5% level of significance to reject the company’s claim that the variance of the amount of fat in the whole milk is no more than 0.25. TRY IT YOURSELF 4 A bottling company claims that the variance of the amount of sports drink in a 12-ounce bottle is no more than 0.40. A random sample of 31 bottles has a variance of 0.75. At a = 0.01, is there enough evidence to reject the company’s claim? Assume the population is normally distributed. Answer: Page A41 EXAMPLE 4 10 20 30 40 50 60 70 0 2 2χ χ α = 55.758 2χ = 43.2 = 0.05
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