Applied Statistics and Probability for Engineers

9-4 HYPOTHESIS TESTS ON THE VARIANCE AND STANDARD DEVIATION OF A NORMAL POPULATION 307

9-4 HYPOTHESIS TESTS ON THE VARIANCE AND STANDARD

DEVIATION OF A NORMAL POPULATION

Sometimes hypothesis tests on the population variance or standard deviation are needed.

When the population is modeled by a normal distribution, the tests and intervals described in

this section are applicable.

9-4.1 The Hypothesis Testing Procedures

Suppose that we wish to test the hypothesis that the variance of a normal population 2 equals

a specified value, say or equivalently, that the standard deviation is equal to 0. Let X 1 ,

X 2 ,p, Xnbe a random sample of nobservations from this population. To test

(9-26)

we will use the test statistic:

H 1 : 2 ^20

H 0 : 2 ^20

20 ,

(b) What is the P-value of the test statistic computed in part (a)?

(c) Compute the power of the test if the true mean dissolved

oxygen concentration is as low as 3.

(d) What sample size would be required to detect a true mean

dissolved oxygen concentration as low as 2.5 if we

wanted the power of the test to be at least 0.9?

9-39. Consider the cigar tar content data first presented in

Exercise 8-82.

(a) Can you support a claim that mean tar content exceeds

1.5? Use     0.05

(b) What is the P-value of the test statistic computed in part (a)?

(c) Compute the power of the test if the true mean tar content

is 1.6.

(d) What sample size would be required to detect a true mean

tar content of 1.6 if we wanted the power of the test to be

at least 0.8?

9-40. Exercise 6-22 gave data on the heights of female

engineering students at ASU.

(a) Can you support a claim that mean height of female engi-

neering students at ASU is 65 inches? Use    0.05

(b) What is the P-value of the test statistic computed in part (a)?

(c) Compute the power of the test if the true mean height is

62 inches.

(d) What sample size would be required to detect a true mean

height of 64 inches if we wanted the power of the test to be

at least 0.8?

9-41. Exercise 6-24 presented data on the concentration of

suspended solids in lake water.

(a) Test the hypotheses H 0 :  55 versus , use

     0.05.

(b) What is the P-value of the test statistic computed in part (a)?

(c) Compute the power of the test if the true mean concentra-

tion is as low as 50.

(d) What sample size would be required to detect a true mean

concentration as low as 50 if we wanted the power of the

test to be at least 0.9?

9-42. Exercise 6-25 describes testing golf balls for an over-

all distance standard.

(a) Can you support a claim that mean distance achieved by

this particular golf ball exceeds 280 yards? Use     0.05.

(b) What is the P-value of the test statistic computed in part (a)?

(c) Compute the power of the test if the true mean distance is

290 yards.

(e) What sample size would be required to detect a true mean

distance of 290 yards if we wanted the power of the test to

be at least 0.8?

H 1 :  55

X^20  (9-27)

1 n 

12 S^2

02

If the null hypothesis is true, the test statistic defined in Equation 9-27

follows the chi-square distribution with n 1 degrees of freedom. This is the reference

H 0 : 2 ^20 X^20

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