Applied Statistics and Probability for Engineers

10-5 INFERENCES ON THE VARIANCES OF TWO NORMAL POPULATIONS 355

10-42. Two different analytical tests can be used to deter-

mine the impurity level in steel alloys. Eight specimens

Subject Before After

1 195 187

2 213 195

3 247 221

4 201 190

5 187 175

6 210 197

7 215 199

8 246 221

9 294 278

10 310 285

are tested using both procedures, and the results are shown in

the following tabulation. Is there sufficient evidence to con-

clude that both tests give the same mean impurity level,

using 0.01?

10-5 INFERENCES ON THE VARIANCES

OF TWO NORMAL POPULATIONS

We now introduce tests and confidence intervals for the two population variances shown in

Fig. 10-1. We will assume that both populations are normal. Both the hypothesis-testing and

confidence interval procedures are relatively sensitive to the normality assumption.

10-5.1 The FDistribution

Suppose that two independent normal populations are of interest, where the population means

and variances, say,  1 , ^21 ,  2 , and ^22 , are unknown. We wish to test hypotheses about the

equality of the two variances, say, H 0 : ^21 ^22. Assume that two random samples of size n 1

from population 1 and of size n 2 from population 2 are available, and let S^21 and S^22 be the sam-

ple variances. We wish to test the hypotheses

(10-24)

The development of a test procedure for these hypotheses requires a new probability

distribution, the Fdistribution. The random variable Fis defined to be the ratio of two

H 1 : ^21 ^22

H 0 : ^21 ^22

(a) Do the data suggest that the two methods prove the same

mean value for natural vibration frequency? Use 

0.05.

(b) Find a 95% confidence interval on the mean difference

between the two methods.

10-41. Ten individuals have participated in a diet-modifica-

tion program to stimulate weight loss. Their weight both be-

fore and after participation in the program is shown in the fol-

lowing list. Is there evidence to support the claim that this

particular diet-modification program is effective in producing

a mean weight reduction? Use 0.05.

10-43. Consider the weight-loss data in Exercise 10-41. Is

there evidence to support the claim that this particular diet-

modification program will result in a mean weight loss of at

least 10 pounds? Use 0.05.

10-44. Consider the weight-loss experiment in Exercise

10-41. Suppose that, if the diet-modification program results

in mean weight loss of at least 10 pounds, it is important to de-

tect this with probability of at least 0.90. Was the use of 10

subjects an adequate sample size? If not, how many subjects

should have been used?

Specimen Test 1 Test 2

1 1.2 1.4

2 1.3 1.7

3 1.5 1.5

4 1.4 1.3

5 1.7 2.0

6 1.8 2.1

7 1.4 1.7

8 1.3 1.6

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