Fundamentals of Probability and Statistics for Engineers

Hence, using the transformation given by Equation (9.123), we have the general
result

This result can also be used to estimate means of nonnormal populations with
known variances if the sample size is large enough to justify use of the central
limit theorem.
It is noteworthy that, in this case, the position of the interval is a function of
X and therefore is a function of the sample. The width of the interval, in
contrast, is a function only of sample size n, being in versely proportional to n1/2.
The [100(1 )] % confidence interval for m given in Equation (9.130) also
provides an estimate of the accuracy of our point estimatorX for m. As we see
from Figure 9.7, the true mean m lies within the indicated interval with
[100(1 )] % confidence. SinceX is at the center of the interval, the distance

fU(u)

1–

Figure 9. 6 [100(1 )]% confidence limits for U

d

X

— m

Figure 9.7 Error in point estimatorXform

Parameter Estimation 297

P…u= 2 <U<u= 2 †ˆ 1 : … 9 : 129 †

P X

u= 2 

n^1 =^2

<m<X‡

u= 2 

n^1 =^2



ˆ 1 : … 9 : 130 †





0

u

α /2 α /2

α

−uα / 2 uα / 2



X – uα / 2 σ

n1/2

X+ uα / 2

σ

n1/2