Section 8.4 Confidence Intervals for a Standard Deviation 391 0 10 20 30 40 50 0.30 0.25 0.20 0.15 0.10 0.05 0 χ2 with 1 degree of freedom χ2 with 10 degrees of freedom χ2 with 20 degrees of freedom χ2 with 5 degrees of freedom Figure 8.13 Area = α/2 χ 2 1−α/2 χα/2 2 Area = 1 − α Area = α/2 Figure 8.14 Confidence intervals for variances and standard deviations follow a somewhat different pattern than those for means and proportions. Whereas confidence intervals for means and proportions consist of a point estimate, a critical value, and a standard error, confidence intervals for variances and standard deviations consist of a point estimate and two critical values. The point estimate for the population variance ��2 is s2, the sample variance. The critical values come from the chi-square distribution. The critical values for a level 100(1 − ��)% confidence interval are the values that contain the middle 100(1 − ��)% of the area under the curve between them. The notation for the critical values tells how much area is to the right of the critical value. Thus, for a level 1 − �� confidence interval, the critical values are denoted ��2 1−��∕2 and ��2 ��∕2. See Figure 8.14. Example 8.19 shows how to use Table A.4 to find critical values. Area = 0.025 EXAMPLE 8.19 Find critical values Area = 0.95 Area = 0.025 χ2 0.975 = 3.247 χ2 0.025 = 20.483 Figure 8.15 Find the critical values for a 95% confidence interval using the chi-square distribution with 10 degrees of freedom. Solution Figure 8.15 presents the chi-square distribution with 10 degrees of freedom and shows the locations of the critical values. The confidence level is 95%, so the critical values are the values that contain the middle 95% of the area under the curve between them. The lower critical value, denoted ��2 0.975, has an area of 0.975 to its right, and the upper critical value, denoted ��2 0.025, has an area of 0.025 to its right. These critical values are found in Table A.4, at the intersection of the row corresponding to 10 degrees of freedom and the columns corresponding to 0.975 and 0.025. The critical values are ��2 0.975 = 3.247 and ��2 0.025 = 20.483. Degrees of Area in Right Tail Freedom 0.995 0.99 0.975 0.95 0.90 0.10 0.05 0.025 0.01 0.005 ⋮ ⋮ ⋮ ⋮ ⋮ ⋮ ⋮ ⋮ ⋮ ⋮ ⋮ 8 1.344 1.646 2.180 2.733 3.490 13.362 15.507 17.535 20.090 21.955 9 1.735 2.088 2.700 3.325 4.168 14.684 16.919 19.023 21.666 23.589 10 2.156 2.558 3.247 3.940 4.865 15.987 18.307 20.483 23.209 25.188 11 2.603 3.053 3.816 4.575 5.578 17.275 19.675 21.920 24.725 26.757 12 3.074 3.571 4.404 5.226 6.304 18.549 21.026 23.337 26.217 28.300 ⋮ ⋮ ⋮ ⋮ ⋮ ⋮ ⋮ ⋮ ⋮ ⋮ ⋮
navidi_monk_elementary_statistics_2e_ch7-9
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