Applied Statistics and Probability for Engineers

344 CHAPTER 10 STATISTICAL INFERENCE FOR TWO SAMPLES

The numerical results from Minitab exactly match the calculations from Example 10-6. Note

that a two-sided 95% CI on  1   2 is also reported. We will discuss its computation in

Section 10-3.4; however, note that the interval does not include zero. Indeed, the upper 95%

of confidence limit is 3.29 ppb, well below zero, and the mean observed difference is

.

10-3.2 More about the Equal Variance Assumption (CD Only)

10-3.3 Choice of Sample Size

The operating characteristic curves in Appendix Charts VIe, VIf, VIg, and VIhare used to

evaluate the type II error for the case where ^21 ^22  ^2. Unfortunately, when ^21  ^22 , the

distribution of is unknown if the null hypothesis is false, and no operating characteristic

curves are available for this case.

For the two-sided alternative H 1 :  1   2     0 , when ^21  ^22  ^2 and n 1  n 2 

n, Charts VIeand VIfare used with

(10-17)

where is the true difference in means that is of interest. To use these curves, they must be

entered with the sample size  2 n 1. For the one-sided alternative hypothesis, we use

Charts VIgand VIhand define dand as in Equation 10-17. It is noted that the parameter d

is a function of , which is unknown. As in the single-sample t-test, we may have to rely on a

prior estimate of or use a subjective estimate. Alternatively, we could define the differences

in the mean that we wish to detect relative to .

EXAMPLE 10-7 Consider the catalyst experiment in Example 10-5. Suppose that, if catalyst 2 produces a mean

yield that differs from the mean yield of catalyst 1 by 4.0%, we would like to reject the null

hypothesis with probability at least 0.85. What sample size is required?

Using sp 2.70 as a rough estimate of the common standard deviation , we have

From Appendix Chart VIewith d0.74 and  

0.15, we find n* 20, approximately. Therefore, since n* 2 n 1,

and we would use sample sizes of n 1  n 2  n 11.

n

n* 1

2



201

2

10.5 111 say 2

dƒƒ
2 ƒ4.0ƒ
31221 2.70 24 0.74.

n*

d

ƒ 0 ƒ

2 

T* 0

x 1 x 2  12  5 17.515 ppb

Two-Sample T-Test and CI: PHX, RuralAZ

Two-sample T for PHX vs RuralAZ

N Mean StDev SE Mean

PHX 10 12.50 7.63 2.4

RuralAZ 10 27.5 15.3 4.9

Difference mu PHX  mu RuralAZ

Estimate for difference: 15.00

95% CI for difference: (26.71, 3.29)

T-Test of difference 0 (vs not ): T-Value 2.77 P-Value 0.016 DF 13

The Minitab output for this example follows:

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