346 CHAPTER 7 Estimating Parameters and Determining Sample Sizes ■ Critical Values of x2 We denote a right-tailed critical value by x2 R and we denote a left-tailed critical value by x2 L. Those critical values can be found by using technology or Table A-4, and they require that we first determine a value for the number of degrees of freedom. ■ Degrees of Freedom For the methods of this section, the number of degrees of freedom is the sample size minus 1. Degrees of freedom: df = n − 1 ■ The chi-square distribution is skewed to the right, unlike the normal and Student t distributions (see Figure 7-7). ■ The values of chi-square can be zero or positive, but they cannot be negative, as shown in Figure 7-7. ■ The chi-square distribution is different for each number of degrees of freedom, as illustrated in Figure 7-8. As the number of degrees of freedom increases, the chi-square distribution approaches a normal distribution. Not symmetric All values are nonnegative 0 x 2 FIGURE 7-7 Chi-Square Distribution 0 5 1015202530354045 x 2 df 5 10 df 5 20 FIGURE 7-8 Chi-Square Distribution for df = 10 and df = 20 CAUTION Because the chi-square distribution is not symmetric, a confidence interval estimate of s2 does not fit a format of s2 - E 6 s2 6 s2 + E, so we must do separate calculations for the upper and lower confidence interval limits. Consequently, a confidence interval can be expressed in a format such as 7.6 6 s 6 14.2 or a format of (7.6, 14.2), but it cannot be expressed in a format of s { E. If using Table A-4 for finding critical values, note the following design feature of that table: In Table A-4, each critical value of X2 in the body of the table corresponds to an area given in the top row of the table, and each area in that top row is a cumulative area to the right of the critical value. CAUTION Table A-2 for the standard normal distribution provides cumulative areas from the left, but Table A-4 for the chi-square distribution uses cumulative areas from the right.
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