Hypothesis Testing for Variance and Standard Deviation 7.5 394 CHAPTER 7 Hypothesis Testing with One Sample What You Should Learn How to find critical values for a chi-square test How to use the chi-square test to test a variance s 2 or a standard deviation s Critical Values for a Chi-SquareTest The Chi-Square Test α α 1 − Critical value 0 2χ 2χ Right-Tailed Test 0 2χ α α 1 − Critical value 2χ Left-Tailed Test 1 2 α α 1 2 α 1 − Critical value L 2χ Critical value R 2χ 2χ Two-Tailed Test Critical Values for a Chi-Square Test In real life, it is important to produce consistent, predictable results. For instance, consider a company that manufactures golf balls. The manufacturer must produce millions of golf balls, each having the same size and the same weight. There is a very low tolerance for variation. For a normally distributed population, you can test the variance and standard deviation of the process using the chi-square distribution with n - 1 degrees of freedom. Before learning how to do the test, you must know how to find the critical values, as shown in the guidelines. Finding Critical Values for a Chi-Square Test 1. Specify the level of significance a. 2. Identify the degrees of freedom, d.f. = n - 1. 3. The critical values for the chi-square distribution are found in Table 6 in Appendix B. To find the critical value(s) for a a. right-tailed test, use the value that corresponds to d.f. and a. b. left-tailed test, use the value that corresponds to d.f. and 1 - a. c. two-tailed test, use the values that correspond to d.f. and 1 2a, and d.f. and 1 - 1 2a. See the figures at the left. GUIDELINES Finding a Critical Value for a Right-Tailed Test Find the critical value x 2 0 for a right-tailed test when n = 26 and a = 0.10. SOLUTION The degrees of freedom are d.f. = n - 1 = 26 - 1 = 25. The figure below shows a chi-square distribution with 25 degrees of freedom and a shaded area of a = 0.10 in the right tail. Using Table 6 in Appendix B with d.f. = 25 and a = 0.10, the critical value is x 2 0 = 34.382. 5 1015202530354045 0 2χ α = 34.382 = 0.10 2χ TRY IT YOURSELF 1 Find the critical value x 2 0 for a right-tailed test when n = 18 and a = 0.01. Answer: Page A41 EXAMPLE 1
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