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Section 8-2/Conidence Interval on the Mean of a Normal Distribution, Variance Known 283

Suppose that the population of interest has a normal distribution with unknown mean μ and
unknown variance σ . Assume that a random sample of size n, say, X , X , …, X , is available,
2
1 2 n
and let X and S be the sample mean and variance, respectively.
2
We wish to construct a two-sided CI on μ. If the variance σ 2 is known, we know that
σ
2
Z = ( X − μ) /( / n) has a standard normal distribution. When σ is unknown, a logical pro-
cedure is to replace σ with the sample standard deviation S. The random variable Z now
becomes T = ( X − μ) /( S n). A logical question is what effect replacing σ with S has on the
distribution of the random variable T. If n is large, the answer to this question is “very little,”
and we can proceed to use the conidence interval based on the normal distribution from Sec-
tion 8-1.5. However, n is usually small in most engineering problems, and in this situation, a
different distribution must be employed to construct the CI.

8-2.1 t DISTRIBUTION

t Distribution
Let X , X , … , X be a random sample from a normal distribution with unknown mean
1 2 n
2
μ and unknown variance σ . The random variable
X − μ
T = (8-13)
S / n
has a t distribution with n – 1 degrees of freedom.

The t probability density function is

Γ[( k + )1 2 ] 1
f x ( ) = ⋅ ( −∞ < x < ∞ (8-14)
/
k ) ⎡
k π Γ( 2 ( 2 ⎤ k+ ) 1 2
⎣ x k) +1 ⎦
where k is the number of degrees of freedom. The mean and variance of the t distribution are
k
zero and k ( − 2 ) (for k > 2), respectively.
Several t distributions are shown in Fig. 8-4. The general appearance of the t distribution
is similar to the standard normal distribution in that both distributions are symmetric and uni-
modal, and the maximum ordinate value is reached when the mean μ = 0. However, the t dis-
tribution has heavier tails than the normal; that is, it has more probability in the tails than does
the normal distribution. As the number of degrees of freedom k → ∞, the limiting form of the
t distribution is the standard normal distribution. Generally, the number of degrees of freedom
for t is the number of degrees of freedom associated with the estimated standard deviation.
Appendix Table V provides percentage points of the t distribution. We will let t be the
α,k
value of the random variable T with k degrees of freedom above which we ind an area (or
probability) α. Thus, t is an upper-tailed 100α percentage point of the t distribution with k
α,k
degrees of freedom. This percentage point is shown in Fig. 8-5. In the Appendix Table V, the
α values are the column headings, and the degrees of freedom are listed in the left column. To
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 = 1.812. That is,
0.05,10
10 ( , ) = ( .
P T >1 812.
P T > t 0 05 10. 10 ) = 0 05
Because the t distribution is symmetric about zero, we have t = –t ; that is, the t-value
1–α,n α,n
having an area of 1 – α to the right (and therefore an area of a to the left) is equal to the nega-
tive of the t-value that has area a in the right tail of the distribution. Therefore, t = –t =
0.95,10 0.05,10
–1.812. Finally, because t ,α ∞ is the standard normal distribution, the familiar z values appear
α
in the last row of Appendix Table V.