Applied Statistics and Probability for Engineers

258 CHAPTER 8 STATISTICAL INTERVALS FOR A SINGLE SAMPLE

t 1 – α,k= – tα,k 0 tα,k t

α α

Section 8-2.5. However, n is usually small in most engineering problems, and in this situation

a different distribution must be employed to construct the CI.

8-3.1 The tDistribution

Let X 1 , X 2 ,p, Xnbe a random sample from a normal distribution with unknown

mean and unknown variance ^2. The random variable

(8-15)

has a tdistribution with n1 degrees of freedom.

T

X

S   1 n

Definition

The tprobability density function is

(8-16)

where kis the number of degrees of freedom. The mean and variance of the tdistribution are

zero and k/(k  2) (for k 2), respectively.

Several tdistributions are shown in Fig. 8-4. The general appearance of the tdistribution is

similar to the standard normal distribution in that both distributions are symmetric and

unimodal, and the maximum ordinate value is reached when the mean   0. However, the t

distribution has heavier tails than the normal; that is, it has more probability in the tails than the

normal distribution. As the number of degrees of freedom ,the limiting form of the tdis-

tribution is the standard normal distribution. Generally, the number of degrees of freedom fort

are the number of degrees of freedom associated with the estimated standard deviation.

Appendix Table IV provides percentage pointsof the tdistribution. We will let t,kbe the

value of the random variable Twith kdegrees of freedom above which we find an area

(or probability) . Thus, t,kis an upper-tail 100percentage point of the tdistribution with k

degrees of freedom. This percentage point is shown in Fig. 8-5. In the Appendix Table IV the

values are the column headings, and the degrees of freedom are listed in the left column. To

kS

f 1 x 2 

 31 k 12  24

2 k 1 k   22



1

31 x^2 k 2  141 k^12  2



x

Figure 8-4 Probability density functions of several t

distributions.

Figure 8-5 Percentage points of the t

distribution.

0

k = ∞ [N (0, 1)]

x

k = 10

k = 1

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