SECTION 6.4 Confidence Intervals for Variance and Standard Deviation 331 χ2 10 20 30 0.025 0.025 0.95 = 30.191 = 7.564 L 2χ R 2χ There are two critical values for each level of confidence. The value x 2 R represents the right-tail critical value and x 2 L represents the left-tail critical value. Table 6 in Appendix B lists critical values of x 2 for various degrees of freedom and areas. Each area listed in the top row of the table represents the region under the chi-square curve to the right of the critical value. Finding Critical Values for X 2 Find the critical values x 2 R and x 2 L for a 95% confidence interval when the sample size is 18. SOLUTION Because the sample size is 18, d.f. = n - 1 = 18 - 1 = 17. Degrees of freedom The area to the right of x 2 R is Area to the right of x 2 R = 1 - c 2 = 1 - 0.95 2 = 0.025 and the area to the right of x 2 L is Area to the right of x 2 L = 1 + c 2 = 1 + 0.95 2 = 0.975. A portion of Table 6 is shown. Using d.f. = 17 and the areas 0.975 and 0.025, you can find the critical values, as shown by the highlighted areas in the table. (Note that the top row in the table lists areas to the right of the critical value. The entries in the table are critical values.) a 0.995 0.99 0.90 0.10 1 — — 0.001 0.004 0.016 2.706 5.024 2 0.010 0.020 0.051 0.103 0.211 4.605 7.378 3 0.072 0.115 0.216 0.352 0.584 6.251 9.348 0.95 0.025 15 4.601 5.229 6.262 7.261 8.547 22.307 27.488 16 5.142 5.812 6.908 7.962 9.312 23.542 28.845 17 5.697 6.408 7.564 8.672 10.085 24.769 30.191 18 6.265 7.015 8.231 9.390 10.865 25.989 31.526 19 6.844 7.633 8.907 10.117 11.651 27.204 32.852 20 7.434 8.260 9.591 10.851 12.443 28.412 34.170 3.841 5.991 7.815 0.05 24.996 26.296 27.587 28.869 30.144 31.410 χ2 L χ2 R Degrees of freedom 0.975 χ2 L χ2 R Area to the right of Area to the right of From the table, you can see that the critical values are x 2 R = 30.191 and x 2 L = 7.564. Interpretation So, for a chi-square distribution curve with 17 degrees of freedom, 95% of the area under the curve lies between 7.564 and 30.191, as shown in the figure at the left. TRY IT YOURSELF 1 Find the critical values x 2 R and x 2 L for a 90% confidence interval when the sample size is 30. Answer: Page A40 EXAMPLE 1 Study Tip For chi-square critical values with a c@confidence level, the values shown below, x 2 L and x 2 R , are what you look up in Table 6 in Appendix B. R 2χ χ2 1 − c 2 Area to the right of x 2 R L 2χ χ2 1 − = ( ( 1 − c 2 1 + c 2 Area to the right of x 2 L The result is that you can conclude that the area between the left and right critical values is c. c L 2χ R 2χ χ2 1 − c 2 1 − c 2
RkJQdWJsaXNoZXIy NjM5ODQ=