Applied Statistics and Probability for Engineers

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

Given the assumptions of this section, the quantity

(10-13)

has a tdistribution with n 1 n 2 2 degrees of freedom.

T

X 1 X 2  1  1  22

Sp

B

1

n 1

1

n 2

The use of this information to test the hypotheses in Equation 10-11 is now straightfor-

ward: simply replace  1  2 by and the resulting test statistichas a tdistribution with

n 1 n 2 2 degrees of freedom under H 0 :  1  2 . Therefore, the reference distribu-

tion for the test statistic is the tdistribution with n 1 n 2 2 degrees of freedom. The location

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.

 0

 0 ,

Null hypothesis: H 0 :  1  2 

Test statistic: (10-14)

Alternative Hypothesis Rejection Criterion

H 1 :  1  2  0 t 0 t ,n 1 n 2  2

H 1 :  1  2 

 0 t 0

 t ,n 1 n 2  2

t 0 t 2,n 1 n 2  2

H 1 :  1  2  0 t 0

 t 2,n 1 n 2  2 or

T 0 

X 1 X 2  0

Sp

B

1

n 1

1

n 2

 0

Definition:

The Two-Sample

or Pooled t-Test*

*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.

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  1 and  2 , the mean process yield using catalysts
   1 and 2, respectively, and we want to know if  1  2 0.


2. H 0 :  1  2 0, or H 0 :  1  2

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