Applied Statistics and Probability for Engineers

One-sided confidence boundson the mean of a normal distribution are also of interest

and are easy to find. Simply use only the appropriate lower or upper confidence limit from

Equation 8-18 and replacet 2,n 1 by t,n 1.

8-3 CONFIDENCE INTERVAL ON THE MEAN OF A NORMAL DISTRIBUTION, VARIANCE UNKNOWN 259

illustrate the use of the table, note that the t-value with 10 degrees of freedom having an area

of 0.05 to the right is t0.05,10  1.812. That is,

Since the tdistribution is symmetric about zero, we have t 1 t; that is, the t-value hav-

ing an area of 1 to the right (and therefore an area of to the left) is equal to the nega-

tive of the t-value that has area in the right tail of the distribution. Therefore, t0.95,10

t0.05,101.812. Finally, because t is the standard normal distribution, the familiar zval-

ues appear in the last row of Appendix Table IV.

8-3.2 Development of the tDistribution (CD Only)

8-3.3 The tConfidence Interval on 

It is easy to find a 100(1 ) percent confidence interval on the mean of a normal distribu-

tion with unknown variance by proceeding essentially as we did in Section 8-2.1. We know

that the distribution of is twith n 1 degrees of freedom. Letting

be the upper 1002 percentage point of the tdistribution with n 1 degrees of

freedom, we may write:

or

Rearranging this last equation yields

(8-17)

This leads to the following definition of the 100(1 ) percent two-sided confidence inter-

val on .

P 1 Xt 2,n 1 S 1 nXt 2,n 1 S 1 n 2  1 

P at 2,n 1 

X

S   1 n

t 2,n 1 b 1 

P 1 t 2,n 1 Tt 2,n 12  1 

t 2,n 1

T 1 X 2 1 S 1 n 2

P 1 T 10

 t0.05,10 2 P 1 T 10

 1.812 2 0.05

If and sare the mean and standard deviation of a random sample from a normal

distribution with unknown variance ^2 , a 100(1) percent confidence interval

on is given by

(8-18)

where is the upper 1002 percentage point of the tdistribution with n 1

degrees of freedom.

t 2,n 1

xt 2,n 1 s 1 nxt 2,n 1 s 1 n

x

Definition

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