Applied Statistics and Probability for Engineers

326 CHAPTER 9 TESTS OF HYPOTHESES FOR A SINGLE SAMPLE

MIND-EXPANDING EXERCISES

In the E-book, click on any

term or concept below to

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Connection between

hypothesis tests

and confidence

intervals

Critical region for a test

statistic

Null hypothesis

One- and two-sided

alternative hypotheses

Operating characteristic

curves

Power of the test

P-value

Reference distribution

for a test statistic

Sample size determina-

tion for hypothesis

tests

Significance level of a

test

Statistical hypotheses

Statistical versus practi-

cal significance

Test for goodness of fit

Test for homogeneity

Test for independence

Test statistic

Type I and type II

errors

CD MATERIAL

Likelihood ratio test

IMPORTANT TERMS AND CONCEPTS

9-91. Suppose that we wish to test H 0 :   0 versus

, where the population is normal with

known . Let , and define the critical region

so that we will reject H 0 if or if

where z 0 is the value of the usual test statistic for these

hypotheses.

(a) Show that the probability of type I error for this test

is .

(b) Suppose that the true mean is. Derive

an expression for for the above test.

9-92. Derive an expression for for the test on the

variance of a normal distribution. Assume that the two-

sided alternative is specified.

9-93. When X 1 , X 2 , p, Xnare independent Poisson

random variables, each with parameter , and nis large,

the sample mean has an approximate normal distribu-

tion with mean and variance. Therefore,

has approximately a standard normal distribution. Thus

we can test H 0 :   0 by replacing in Zby  0. When Xi

are Poisson variables, this test is preferable to the large-

sample test of Section 9-2.5, which would use in

the denominator, because it is designed just for the

Poisson distribution. Suppose that the number of open cir-

cuits on a semiconductor wafer has a Poisson distribution.

Test data for 500 wafers indicate a total of 1038 opens.

Using  0.05, does this suggest that the mean number

of open circuits per wafer exceeds 2.0?

S 1 n

Z

X 

1 n

n

X

 1  0

z 0 z z 0  z 

,

0 



H 1 :  0

9-94. When X 1 , X 2 , p, Xnis a random sample from a

normal distribution and nis large, the sample standard

deviation has approximately a normal distribution with

mean and variance. Therefore, a large-sample

test for H 0 :   0 can be based on the statistic

Use this result to test H 0 :  10 versus H 1 :  10 for

the golf ball overall distance data in Exercise 6-25.

9-95. Continuation of Exercise 9-94.Using the

results of the previous exercise, find an approximately

unbiased estimator of the 95 percentile  

 1.645.

From the fact that and Sare independent random

variables, find the standard error of . How would you

estimate the standard error?

9-96. Continuation of Exercises 9-94 and 9-95.

Consider the golf ball overall distance data in Exercise

6-25. We wish to investigate a claim that the 95 per-

centile of overall distance does not exceed 285 yards.

Construct a test statistic that can be used for testing the

appropriate hypotheses. Apply this procedure to the data

from Exercise 6-25. What are your conclusions?

9-97. Let X 1 , X 2 , p, Xnbe a sample from an exponen-

tial distribution with parameter . It can be shown that

has a chi-square distribution with 2ndegrees

of freedom. Use this fact to devise a test statistic and

critical region for H 0 :   0 versus the three usual

alternatives.

2  ni 1 Xi

X

Z

S    0

2 ^20  12 n 2

^2  12 n 2

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