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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 t 0.05,10 1.812. That is,

P1T
t 0.05,10 2 P1T
1.8122 0.05
10
10
t ; that is, the t-value hav-
Since the t distribution is symmetric about zero, we have t 1
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, t 0.95,10
t 0.05,10 1.812. Finally, because t is the standard normal distribution, the familiar z val-

ues appear in the last row of Appendix Table IV.
8-3.2 Development of the t Distribution (CD Only)

8-3.3 The t Confidence 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 T 1X 2 1S 1n2 is t with n 1 degrees of freedom. Letting
t 2,n 1 be the upper 100 2 percentage point of the t distribution with n 1 degrees of
freedom, we may write:

P 1 t 2,n 1 T t 2,n 1 2 1

or
X
P a t 2,n 1 t 2,n 1 b 1
S 1n

Rearranging this last equation yields

P 1X t S 1n X t S 1n2 1 (8-17)
2,n 1 2,n 1
This leads to the following definition of the 100(1 ) percent two-sided confidence inter-
val on .

Definition
If x and s are the mean and standard deviation of a random sample from a normal
2
distribution with unknown variance , a 100(1 ) percent confidence interval
on is given by

x t 2,n 1 s 1n x t 2,n 1 s 1n (8-18)

is the upper 100 2 percentage point of the t distribution with n 1
where t 2,n 1
degrees of freedom.

One-sided confidence bounds on 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 replace t 2,n 1 by t ,n 1 .