Page 315 - Fundamentals of Probability and Statistics for Engineers

298 Fundamentals of Probability and Statistics for Engineers

between X and m can be at most equal to one-half of the interval width. We
thus have the result given in Theorem 9.6.
Theorem 9.6: let X be an estimator for m. Then, with [100(1 )]% con-
fidence, the error of using this estimator for m is less than
u /2

n 1/2

Example 9.18. Problem: let population X be normally distributed with

known variance 2 .If X is used as an estimator for mean m, determine the
sample size n needed so that the estimation error will be less than a specified
amount " with [100(1 )] % confidence.
Answer: using the theorem given above, the minimum sample size n must
satisfy
u =2
" ˆ :
n 1=2
Hence, the solution for n is
u =2 2

n ˆ : 9:131†
"

9.3.2.2 Confidence Interval for m in N(m, s 2 ) with Unknown s 2

The difference between this problem and the preceding one is that, since is not
known, we can no longer use

1
U ˆ X m†
n 1=2
as the random variable for confidence limit calculations regarding mean m. Let
2
us then use sample variance S as an unbiased estimator for 2 and consider the
random variable
S
1
Y ˆ X m† : 9:132†
n 1=2
The random variable Y is now a function of random variables X and S. In
order to determine its distribution, we first state Theorem 9.7.

Theorem 9.7: Student ’s t-distribution. Consider a random variable T defined by

1=2

V
T ˆ U : 9:133†
n
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