Applied Statistics and Probability for Engineers

10-3 INFERENCE FOR THE DIFFERENCE IN MEANS OF TWO NORMAL DISTRIBUTIONS, VARIANCES UNKNOWN 341

The Minitab two-sample t-test and confidence interval procedure for Example 10-5

follows:

Notice that the numerical results are essentially the same as the manual computations in

Example 10-5. The P-value is reported as P 0.73. The two-sided CI on  1   2 is also

reported. We will give the computing formula for the CI in Section 10-3.3. Figure 10-2 shows

the normal probability plot of the two samples of yield data and comparative box plots. The

normal probability plots indicate that there is no problem with the normality assumption.

Furthermore, both straight lines have similar slopes, providing some verification of the as-

sumption of equal variances. The comparative box plots indicate that there is no obvious dif-

ference in the two catalysts, although catalyst 2 has slightly greater sample variability.

Case 2: ^21 ^22

In some situations, we cannot reasonably assume that the unknown variances ^21 and ^22 are

equal. There is not an exact t-statistic available for testing H 0 :  1   2   0 in this case.

However, if H 0 :  1   2   0 is true, the statistic

Two-Sample T-Test and CI: Cat 1, Cat 2

Two-sample T for Cat 1 vs Cat 2

N Mean StDev SE Mean

Cat 1 8 92.26 2.39 0.84

Cat 2 8 92.73 2.99 1.1

Difference mu Cat 1  mu Cat 2

Estimate for difference: 0.48

95% CI for difference: (3.37, 2.42)

T-Test of difference 0 (vs not ): T-Value 0.35 P-Value 0.730 DF 14

Both use Pooled StDev 2.70

1

88

5

10

20

30

40

50

60

70

80

90

95

99

Percentage

93 98

Cat 1

Cat 2

Yield data

(a)(b)

1

90

92

94

96

98

88

2

Yield

Catalyst type

Figure 10-2 Normal probability plot and comparative box plot for the catalyst yield data in Example 10-5.

(a) Normal probability plot, (b) Box plots.

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