Fundamentals of Probability and Statistics for Engineers

Returning to random variable Y defined by Equation (9.132), let

and

Then

where U is clearly distributed according to N(0,1). We also see from Section
9.1.2 that (n 1)S^2 /^2 has the chi-squared distribution with (n 1) degrees of
freedom. Furthermore, although we will not verify it here, it can be shown that
XandS^2 are independent. In accordance with Theorem 9.7, random variable Y
thus has a t-distribution with (n 1) degrees of fr eedom.
The random variable Y can now be used to establish confidence intervals for
mean m. We note that the value of Y depends on the unknown mean m, but its
distribution does not.
The t-distribution is tabulated in Table A.4 in Appendix A. Let tn, /2 be the
value such that

with n representing the number of degrees of fr eedom (see Figure 9.8). We have
the result

U pon substituting Equation (9.132) into Equation
confidence interval for mean m is thus given by

Since bothX and S are functions of the sample, both the position and the width
of the confidence interval given above will vary from sample to sample.

300 Fundamentals of Probability and Statistics for Engineers

Uˆ…Xm†



n^1 =^2

 1

Vˆ

…n 1 †S^2

^2

YˆU

V

n 1

 1 = 2

; … 9 : 139 †

  



P…T>tn;= 2 †ˆ

2

;

P…tn 1 ;= 2 <Y<tn 1 ;= 2 †ˆ 1 : … 9 : 140 †

9.140), a [1001 )]%

P X

tn 1 ;= 2 S

n^1 =^2

<m<X‡

tn 1 ;= 2 S

n^1 =^2



ˆ 1 : … 9 : 141 †