Fundamentals of Probability and Statistics for Engineers

number of degrees of freedom can be determined by interpolation from tabu-
lated values of the PDF of the chi-squared distribution.
Table A.5 in Appendix A gives selected values of^2 n, for various values of n
and. F or convenience, Equation (9.144) is commonly used for constructing
two-sided confidence intervals for^2 of a normal population. If a one-sided
confidence interval is desired, it is then given by (see F igure 9.10)

Example 9.20.Consider Example 9.19 again; let us determine both two-sided
and one-sided 95% confidence intervals for^2.
As seen from Example 9.19, the observed sample variance s^2 ,is

The values of are obtained from Table A.5 to be as
follows:

Equations (9.144) and (9.145) thus lead to, with n 10 and 0.05,

fD(d)

1–

d

n,

2

Figure 9. 10 One-sided [100(1 )]% confidence limit for D with n degrees of freedom

Parameter Estimation 303

α

α

χ α





P ^2 >

…n 1 †S^2

^2 n 1 ;

"

ˆ 1 : … 9 : 145 †



s^2 ˆ 1414 : 3 :

^2 9, 0: 975 ,^2 9, 0: 025 , and^2 9, 0: 05

^29 ; 0 : 975 ˆ 2 : 700 ;^29 ; 0 : 025 ˆ 19 : 023 ;^29 ; 0 : 05 ˆ 16 : 919 :

ˆ ˆ

P… 669 : 12 <^2 < 4714 : 33 †ˆ 0 : 95 ;