Page 395 - Applied Statistics And Probability For Engineers

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10-3 INFERENCE FOR THE DIFFERENCE IN MEANS OF TWO NORMAL DISTRIBUTIONS, VARIANCES UNKNOWN 339

Given the assumptions of this section, the quantity

X X 1 2
1
1
2
2
T (10-13)
1 1
S p n n
B 1 2
n 2 degrees of freedom.
has a t distribution with n 1 2

The use of this information to test the hypotheses in Equation 10-11 is now straightfor-
ward: simply replace by , and the resulting test statistic has a t distribution with
0
1
2
. Therefore, the reference distribu-
n n 2 degrees of freedom under H : 0
1
0
2
1
2
tion for the test statistic is the t distribution with n n 2 degrees of freedom. The location
2
1
of the critical region for both two- and one-sided alternatives parallels those in the one-sample
case. Because a pooled estimate of variance is used, the procedure is often called the pooled
t-test.
Definition:
The Two-Sample Null hypothesis: H 0 : 1 2 0
or Pooled t-Test*
X X 0
2
1
Test statistic: T (10-14)
0
1 1

S p
n
B 1 n 2
Alternative Hypothesis Rejection Criterion
H : 0 t
t
2,n 1 n 2 2 or
0
2
1
1
t t
2,n 1 n 2 2
0
H :
0 t
t ,n 1 n 2 2
1
2
1
0
H : 0 t 0 t ,n 1 n 2 2
2
1
1
EXAMPLE 10-5 Two catalysts are being analyzed to determine how they affect the mean yield of a chemical
process. Specifically, catalyst 1 is currently in use, but catalyst 2 is acceptable. Since catalyst
2 is cheaper, it should be adopted, providing it does not change the process yield. A test is run
in the pilot plant and results in the data shown in Table 10-1. Is there any difference between
the mean yields? Use 0.05, and assume equal variances.
The solution using the eight-step hypothesis-testing procedure is as follows:
1. The parameters of interest are and , the mean process yield using catalysts
2
1
1 and 2, respectively, and we want to know if 0.
2
1
2. H : 0, or H : 2
2
0
1
0
1
*While we have given the development of this procedure for the case where the sample sizes could be different, there
is an advantage to using equal sample sizes n 1 n 2 n. When the sample sizes are the same from both populations,
the t-test is more robust to the assumption of equal variances. Please see Section 10-3.2 on the CD.