SECTION 7.5 Hypothesis Testing for Variance and Standard Deviation 395 Finding a Critical Value for a Left-Tailed Test Find the critical value x 2 0 for a left-tailed test when n = 11 and a = 0.01. SOLUTION The degrees of freedom are d.f. = n - 1 = 11 - 1 = 10. The figure at the left shows a chi-square distribution with 10 degrees of freedom and a shaded area of a = 0.01 in the left tail. The area to the right of the critical value is 1 - a = 1 - 0.01 = 0.99. Using Table 6 with d.f. = 10 and the area 0.99, the critical value is x 2 0 = 2.558. You can check your answer using technology, as shown below. MINITAB Inverse Cumulative Distribution Function Chi-Square with 10 DF P ( X … x) x 0.01 2.55821 TRY IT YOURSELF 2 Find the critical value x 2 0 for a left-tailed test when n = 30 and a = 0.05. Answer: Page A41 Note that because chi-square distributions are not symmetric (like normal or t-distributions), in a two-tailed test the two critical values are not opposites. Each critical value must be calculated separately, as shown in the next example. Finding Critical Values for a Two-Tailed Test Find the critical values x 2 L and x 2 R for a two-tailed test when n = 9 and a = 0.05. SOLUTION The degrees of freedom are d.f. = n - 1 = 9 - 1 = 8. The figure shows a chi-square distribution with 8 degrees of freedom and a shaded area of 1 2a = 0.025 in each tail. The area to the right of x 2 R is 1 2a = 0.025, and the area to the right of x 2 L is 1 - 1 2a = 0.975. Using Table 6 with d.f. = 8 and the areas 0.025 and 0.975, the critical values are x 2 R = 17.535 and x 2 L = 2.180. You can check you answer using technology, as shown at the left. TRY IT YOURSELF 3 Find the critical values x 2 L and x 2 R for a two-tailed test when n = 51 and a = 0.01. Answer: Page A41 EXAMPLE 2 20 15 10 5 1 2 L 2χ α = 2.180 R 2χ = 17.535 = 0.025 1 2 α= 0.025 2χ 20 15 10 5 0 2 2 χ α = 2.558 = 0.01 χ EXCEL 17.53454614 =CHISQ.INV(0.025,8) =CHISQ.INV.RT(0.025,8) A 1 2 2.179730747 EXAMPLE 3
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