7-3 Estimating a Population Standard Deviation or Variance 347 0 x 2 L 5 10.283 x 2 R 5 35.479 x 2 (df 5 21) 0.025 0.025 Table A-4: Use df 5 21 and a cumulative right area of 0.975. Table A-4: Use df 5 21 and a cumulative right area of 0.025. FIGURE 7-9 Finding Critical Values of X2 The critical value to the right (x2 R = 35.479) is obtained from Table A-4 in a straightforward manner by locating 21 in the degrees-of-freedom column at the left and 0.025 across the top row. The leftmost critical value of x2 L = 10.283 also corresponds to 21 in the degrees-of-freedom column, but we must locate 0.975 (or 1 - 0.025) across the top row because the values in the top row are always areas to the right of the critical value. Refer to Figure 7-9 and see that the total area to the right of x2 L = 10.283 is 0.975. YOUR TURN. Find the critical values in Exercise 5 “Nicotine in Menthol Cigarettes.” When obtaining critical values of x2 from Table A-4, if a number of degrees of freedom is not found in the table, you can be conservative by using the next lower number of degrees of freedom, or you can use the closest critical value in the table, or you can get an approximate result with interpolation. For numbers of degrees of freedom greater than 100, use the equation given in Exercise 23 “Finding Critical Values” on pages 354–355, or use a more extensive table, or use technology. Although s2 is the best point estimate of s 2, there is no indication of how good it is, so we use a confidence interval that gives us a range of values associated with a confidence level. EXAMPLE 1 Finding Critical Values of X2 A simple random sample of 22 pulse rates is obtained (as in Example 2, which follows). Construction of a confidence interval for the population standard deviation s requires the left and right critical values of x2 corresponding to a confidence level of 95% and a sample size of n = 22. Find x2 L (the critical value of x2 separating an area of 0.025 in the left tail), and find x2 R (the critical value of x2 separating an area of 0.025 in the right tail). SOLUTION With a sample size of n = 22, the number of degrees of freedom is df = n - 1 = 21. See Figure 7-9.
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