Applied Statistics and Probability for Engineers

330 CHAPTER 10 STATISTICAL INFERENCE FOR TWO SAMPLES

EXAMPLE 10-1 A product developer is interested in reducing the drying time of a primer paint. Two formula-

tions of the paint are tested; formulation 1 is the standard chemistry, and formulation 2 has a

new drying ingredient that should reduce the drying time. From experience, it is known that

the standard deviation of drying time is 8 minutes, and this inherent variability should be un-

affected by the addition of the new ingredient. Ten specimens are painted with formulation 1,

and another 10 specimens are painted with formulation 2; the 20 specimens are painted in

random order. The two sample average drying times are minutes and

minutes, respectively. What conclusions can the product developer draw about the effective-

ness of the new ingredient, using 0.05?

We apply the eight-step procedure to this problem as follows:

1. The quantity of interest is the difference in mean drying times,  1  2 , and  0 0.


2. 
   


3. We want to reject H 0 if the new ingredient reduces mean drying time.


4. 0.05


5. The test statistic is

where ^21 ^22 64 and n 1 n 2 10.

6. Reject H 0 :  1  2 if z 0

 1.645z0.05.

7. Computations: Since minutes and minutes, the test statistic is

z 0 

121  112

B

1822

10

1822

10

2.52

x 1  121 x 2  112

1822

z ̨ 0 

x 1 x 2  0

B

^21

n 1

^22

n 2

H ̨ 1 :  1 

 2.

H ̨ 0 :  1  2 0, or H ̨ 0 : ̨  1  2.

x ̨ 1  121 x ̨ 2  112

distribution when H 0 is true, we would takez and z as the boundaries of the critical re-

gion just as we did in the single-sample hypothesis-testing problem of Section 9-2.1. This

would give a test with level of significance. Critical regions for the one-sided alternatives

would be located similarly. Formally, we summarize these results below.



2 

2

Null hypothesis:

Test statistic: (10-2)

Alternative Hypotheses Rejection Criterion

H ̨ 1 :  1  2  0 z 0 z

H ̨ 1 :  1  2 

 0 z 0

 z

H 1 ̨:  1  2  0 z 0

 z 

2 or z 0 z 

2

Z 0 

X 1 X 2  0

B

^21

n 1

^22

n 2

H 0 :  1  2  0

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