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196 CHAPTER 6 ESTIMATION
FIGURE 6.9.1 Chi-square distributions for several values of degrees of freedom k.
(Source: Paul G. Hoel and Raymond J. Jessen, Basic Statistics for Business and
Economics, Wiley, 1971. Used with permission.)
designate these two values of as x 2 a>2 and x 2 1-1a>22, respectively. The 10011 - a2 percent
confidence interval for 1n - 12s 2 >s 2 , then, is given by
s 2
We now manipulate this expression in such a way that we obtain an expression with
alone as the middle term. First, let us divide each term by 1n - 12s 2 to get
x 2 a>2
1n - 12s 2 6 1 s 2 6 x2 1- 1a>22
1n - 12s 2
If we take the reciprocal of this expression, we have
x 2
x 2 a>2 6
1n - 12 s2
s 2 6 x 2 1- 1a>22
1n - 12s 2
x 2 a>2
7 s 2 7
1n - 12s
2
x 2 1- 1a>22
Note that the direction of the inequalities changed when we took the reciprocals. If we
reverse the order of the terms, we have
1n - 12s 2
x 2 1 - 1a>22
6 s 2 6
1n - 12s2
x 2 a>2
(6.9.1)