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

(b) What is the P-value of the test statistic computed in part (a)? (d) What sample size would be required to detect a true mean
(c) Compute the power of the test if the true mean dissolved height of 64 inches if we wanted the power of the test to be
oxygen concentration is as low as 3. at least 0.8?
(d) What sample size would be required to detect a true mean 9-41. Exercise 6-24 presented data on the concentration of
dissolved oxygen concentration as low as 2.5 if we suspended solids in lake water.
wanted the power of the test to be at least 0.9? (a) Test the hypotheses H 0 : 55 versus H 1 : 55 , use
9-39. Consider the cigar tar content data first presented in 0.05.
Exercise 8-82. (b) What is the P-value of the test statistic computed in part (a)?
(a) Can you support a claim that mean tar content exceeds (c) Compute the power of the test if the true mean concentra-
1.5? Use 0.05 tion is as low as 50.
(b) What is the P-value of the test statistic computed in part (a)? (d) What sample size would be required to detect a true mean
(c) Compute the power of the test if the true mean tar content concentration as low as 50 if we wanted the power of the
is 1.6. test to be at least 0.9?
(d) What sample size would be required to detect a true mean 9-42. Exercise 6-25 describes testing golf balls for an over-
tar content of 1.6 if we wanted the power of the test to be all distance standard.
at least 0.8? (a) Can you support a claim that mean distance achieved by
9-40. Exercise 6-22 gave data on the heights of female this particular golf ball exceeds 280 yards? Use 0.05.
engineering students at ASU. (b) What is the P-value of the test statistic computed in part (a)?
(a) Can you support a claim that mean height of female engi- (c) Compute the power of the test if the true mean distance is
neering students at ASU is 65 inches? Use 0.05 290 yards.
(b) What is the P-value of the test statistic computed in part (a)? (e) What sample size would be required to detect a true mean
(c) Compute the power of the test if the true mean height is distance of 290 yards if we wanted the power of the test to
62 inches. be at least 0.8?

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

2
Suppose that we wish to test the hypothesis that the variance of a normal population equals
2
a specified value, say 0 , or equivalently, that the standard deviation is equal to 0 . Let X 1 ,
X 2 , p , X n be a random sample of n observations from this population. To test
2 2
0
H : 0
(9-26)
2 2
H 1 : 0
we will use the test statistic:

1n
12S 2
2
X 0 (9-27)
2
0
2 2 2
If the null hypothesis H 0 : 0 is true, the test statistic X 0 defined in Equation 9-27
follows the chi-square distribution with n
1 degrees of freedom. This is the reference