Fundamentals of Probability and Statistics for Engineers

betweenX 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:letX be an estimator for m. Then, with [100(1 )]% con-
fidence, the error of using this estimator for m is less than

u/2

n1/2

Ex ample 9. 18. Problem: let population X be normally distributed with
known variance^2 .IfX 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

Hence, the solution for n is

9.3.2.2 Confidence Interval formin N(m^2 )with Unknown^2

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

as the random variable for confiden ce limit calculations regarding mean m. Let
us then use sample variance S^2 as an unbiased estimator for^2 and consider the
random variable

The random variable Y is now a function of random variablesX and S. In
order to determine its distribution, we first state Theorem 9.7.

Theo re m 9. 7 : Student ’s t-dist ribut ion. Consider a random variable T defined by

298 Fundamentals of Probability and Statistics for Engineers







" 

"ˆ

u= 2 

n^1 =^2

:

nˆ

u= 2 

"

 2

: … 9 : 131 †

,s s



Uˆ…Xm†



n^1 =^2

 1



Y ˆ…Xm†

S

n^1 =^2

 1

:… 9 : 132 †

TˆU

V

n

 1 = 2

: … 9 : 133 †