Introduction to Probability and Statistics for Engineers and Scientists

286 Chapter 7: Parameter Estimation

whereas a sample of 10 specimens of the second yielded the data

Type 2: 3,094, 3,106, 3,004, 3,066, 2,984,

3,124, 3,316, 3,212, 3,380, 3,018

If we assume that the samples are normal with a common variance, determine

(a) a 95 percent two-sided confidence interval forμ 1 −μ 2 , the difference in

means;

(b) a 95 percent one-sided upper confidence interval forμ 1 −μ 2 ;

(c) a 95 percent one-sided lower confidence interval forμ 1 −μ 2.

42.Independent random samples are taken from the output of two machines on

a production line. The weight of each item is of interest. From the first machine,

a sample of size 36 is taken, with sample mean weight of 120 grams and a sam-

ple variance of 4. From the second machine, a sample of size 64 is taken, with

a sample mean weight of 130 grams and a sample variance of 5. It is assumed that

the weights of items from the first machine are normally distributed with mean

μ 1 and varianceσ^2 , and that the weights of items from the second machine are

normally distributed with meanμ 2 and varianceσ^2 (that is, the variances are

assumed to be equal). Find a 99 percent confidence interval forμ 1 −μ 2 , the

difference in population means.

43.Do Problem 42 when it is known in advance that the population variances are

4 and 5.

44.The following are the burning times in seconds of floating smoke pots of two

different types.

Type I Type II

481 572 526 537

506 561 511 582

527 501 556 605

661 487 542 558

501 524 491 578

Find a 99 percent confidence interval for the mean difference in burning times

assuming normality with unknown but equal variances.

45.IfX 1 ,...,Xnis a sample from a normal population having known meanμ 1

and unknown varianceσ 12 , andY 1 ,...,Ymis an independent sample from

a normal population having known mean μ 2 and unknown variance σ 22 ,

determine a 100(1−α) percent confidence interval forσ 12 /σ 22.

46.Two analysts took repeated readings on the hardness of city water. Assuming

that the readings of analysti constitute a sample from a normal population