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6.7 Test of Hypothesis about a Population Variance (Optional)

Although many practical problems involve inferences about a population mean (or proportion), it is sometimes of interest to make an inference about a population variance $σ^{2} .$ $σ^{2} .$ To illustrate, a quality control supervisor in a cannery knows that the exact amount each can contains will vary, since there are certain uncontrollable factors that affect the amount of fill. The mean fill per can is important, but equally important is the variance of the fill. If $σ^{2}$ $σ^{2}$ is large, some cans will contain too little and others too much. Suppose regulatory agencies specify that the standard deviation of the amount of fill in 16-ounce cans should be less than .1 ounce. To determine whether the process is meeting this specification, the supervisor randomly selects 10 cans and weighs the contents of each. The results are given in Table 6.6.

Alternate View

16.00 15.95 16.10 16.02 15.99

16.06 16.04 16.05 16.03 16.02

Data Set: FILLWT

Do these data provide sufficient evidence to indicate that the variability is as small as desired? To answer this question, we need a procedure for testing a hypothesis about $σ^{2} .$ $σ^{2} .$

Intuitively, it seems that we should compare the sample variance $σ^{2}$ $σ^{2}$ with the hypothesized value of $σ^{2}$ $σ^{2}$ (or s with $σ$ $σ$ ) in order to make a decision about the population’s variability. The quantity

\frac{(n - 1) s^{2}}{σ^{2}}

$\frac{(n - 1) s^{2}}{σ^{2}}$

is known to have a chi-square $(χ^{2})$ $(χ^{2})$ distribution when the population from which the sample is taken is normally distributed. (Chi-square distributions were introduced in optional Section 5.6.)

Since the distribution of $\frac{(n - 1) s^{2}}{σ^{2}}$ $\frac{(n - 1) s^{2}}{σ^{2}}$ is known, we can use this quantity as a test statistic in a test of hypothesis for a population variance, as illustrated in the next example.

Biography Friedrich R. Helmert (1843–1917)

Helmert Transformations

German Friedrich Helmert studied engineering sciences and mathematics at Dresden University, where he earned his Ph.D. Then he accepted a position as a professor of geodesy—the scientific study of the earth’s size and shape—at the technical school in Aachen. Helmert’s mathematical solutions to geodesy problems led him to several statistics-related discoveries. His greatest statistical contribution occurred in 1876, when he was the first to prove that the sampling distribution of the sample variance $s^{2}$ $s^{2}$ is a chi-square distribution. Helmert used a series of mathematical transformations to obtain the distribution of $s^{2}$ $s^{2}$ —transformations that have since been named “Helmert transformations” in his honor. Later in life, Helmert was appointed professor of advanced geodesy at the prestigious University of Berlin and director of the Prussian Geodetic Institute.

FILLWT Example 6.12 A One-Tailed Test for $σ^{2}$ $σ^{2}$ —Fill Weight Variance

Problem

Refer to the fill weights for the sample of ten 16-ounce cans in Table 6.6. Is there sufficient evidence to conclude that the true standard deviation $σ$ $σ$ of the fill measurements of 16-ounce cans is less than .1 ounce?

Solution

Here, we want to test whether $σ < .1 .$ $σ < .1 .$ Since the null and alternative hypotheses must be stated in terms of $σ^{2}$ $σ^{2}$ (rather than $σ$ $σ$ ), we want to test the null hypothesis that $σ^{2} = (.1)^{2} = .01$ $σ^{2} = (.1)^{2} = .01$ against the alternative that $σ^{2} < .01 .$ $σ^{2} < .01 .$ Therefore, the elements of the test are

$\begin{array}{l} H_{0} : σ^{2} = .01 (Fill variance equals .01 — i .e ., process specifications are not met .) \\ H_{a} : σ^{2} < .01 (Fill variance is less than .01 — i .e ., process specifications are met .) \\ T e s t s t a t i s t i c : χ^{2} = \frac{(n - 1) s^{2}}{σ^{2}} \end{array}$ $\begin{array}{l} H_{0} : σ^{2} = .01 (Fill variance equals .01 — i .e ., process specifications are not met .) \\ H_{a} : σ^{2} < .01 (Fill variance is less than .01 — i .e ., process specifications are met .) \\ T e s t s t a t i s t i c : χ^{2} = \frac{(n - 1) s^{2}}{σ^{2}} \end{array}$
- Assumption: The distribution of the amounts of fill is approximately normal.
- Rejection region: The smaller the value of s² we observe, the stronger is the evidence in favor of H_a. Thus, we reject H₀ for “small values” of the test statistic. Recall that the chi-square distribution depends on $(n - 1)$ $(n - 1)$ degrees of freedom. With $α = .05$ $α = .05$ and $(n - 1) = 9$ $(n - 1) = 9$ df, the critical $χ^{2}$ $χ^{2}$ value for rejection is found in Table IV of Appendix B, and pictured in Figure 6.24: Reject H₀ if $χ^{2} < 3.32511$ $χ^{2} < 3.32511$ .
[Note: The area given in Table IV is the area to the right of the numerical $χ^{2}$ $χ^{2}$ value. Thus, to determine the lower-tail value, which has $α = .05$ $α = .05$ to its left, we use the $χ_{.95}^{2}$ $χ_{.95}^{2}$ column in Table IV.]

A SAS printout of the analysis is displayed in Figure 6.24. The value of s (highlighted on the printout) is $s = .0412 .$ $s = .0412 .$ Substituting into the formula for the test statistic, we have

$χ^{2} = \frac{(n - 1) s^{2}}{σ^{2}} = \frac{9 (.0412)^{2}}{.01} = 1.53$ $χ^{2} = \frac{(n - 1) s^{2}}{σ^{2}} = \frac{9 (.0412)^{2}}{.01} = 1.53$

Figure 6.24 Rejection region for Example 6.12

Figure 6.25 SAS test of fill amount variance, Example 6.12

Conclusion: Since the value of the test statistic is less than 3.32511, the supervisor can conclude that the variance $σ^{2}$ $σ^{2}$ of the population of all amounts of fill is less than .01 (i.e., $σ < .1$ $σ < .1$ ), with probability of a Type I error equal to $α = .05 .$ $α = .05 .$ If this procedure is repeatedly used, it will incorrectly reject $H_{0}$ $H_{0}$ only 5% of the time. Thus, the quality control supervisor is confident in the decision that the cannery is operating within the desired limits of variability.

Look Back

Note that both the test statistic and the one-tailed p-value of the test are shown on the SAS printout. Note also that the p-value (.003) is less than $α = .05,$ $α = .05,$ thus confirming our conclusion to reject $H_{0} .$ $H_{0} .$

Now Work Exercise 6.107

One-tailed and two-tailed tests of hypothesis for $σ^{2}$ $σ^{2}$ are given in the following box.

Large-Sample Test of Hypothesis about $σ^{2 :}$ $σ^{2 :}$ Chi-Square Statistic

Test statistic: $χ_{c =}^{2} \frac{(n - 1) s^{2}}{σ_{0}^{2}}$ $χ_{c =}^{2} \frac{(n - 1) s^{2}}{σ_{0}^{2}}$
One-Tailed Tests		Two-Tailed Test
$H_{0} : σ^{2} = σ_{0}^{2}$ $H_{0} : σ^{2} = σ_{0}^{2}$	$H_{0} : σ^{2} = σ_{0}^{2}$ $H_{0} : σ^{2} = σ_{0}^{2}$	$H_{0} : σ^{2} = σ_{0}^{2}$ $H_{0} : σ^{2} = σ_{0}^{2}$
$H_{a} : σ^{2} < σ_{0}^{2}$ $H_{a} : σ^{2} < σ_{0}^{2}$	$H_{a} : σ^{2} > σ_{0}^{2}$ $H_{a} : σ^{2} > σ_{0}^{2}$	$H_{a} : σ^{2} \neq σ_{0}^{2}$ $H_{a} : σ^{2} \neq σ_{0}^{2}$
Rejection region: $χ_{c}^{2} < χ^{2}_{(1 - α)}$ $χ_{c}^{2} < χ^{2}_{(1 - α)}$	$χ_{c}^{2} > χ_{α}^{2}$ $χ_{c}^{2} > χ_{α}^{2}$	$χ_{c}^{2} < χ_{(1 - α / 2)}^{2} or χ_{c}^{2} > χ_{α / 2}^{2}$ $χ_{c}^{2} < χ_{(1 - α / 2)}^{2} or χ_{c}^{2} > χ_{α / 2}^{2}$
p-value: $P (χ^{2} < χ_{c}^{2})$ $P (χ^{2} < χ_{c}^{2})$	$P (χ^{2} > χ_{c}^{2})$ $P (χ^{2} > χ_{c}^{2})$	$2 \min {(P χ^{2} < χ_{c}^{2}), P (χ^{2} > χ_{c}^{2})}$ $2 \min {(P χ^{2} < χ_{c}^{2}), P (χ^{2} > χ_{c}^{2})}$
Decision: Reject `H`₀ if $α > p - v a l u e$ $α > p - v a l u e$ or if test statistic $(χ_{c)}^{2}$ $(χ_{c)}^{2}$ falls in rejection region where $P (χ^{2} > χ_{α}^{2}) = α, P (χ^{2} < χ_{(1 - α)}^{2}) = α, α = P (Type I error) = P (Reject H_{0} \| H_{0} true)$ $P (χ^{2} > χ_{α}^{2}) = α, P (χ^{2} < χ_{(1 - α)}^{2}) = α, α = P (Type I error) = P (Reject H_{0} \| H_{0} true)$ and the chi-square distribution is based on $(n - 1)$ $(n - 1)$ degrees of freedom.

[Note: The symbol for the numerical value assigned to $σ^{2}$ $σ^{2}$ under the null hypothesis is $σ_{0}^{2}$ $σ_{0}^{2}$ .]

Conditions Required for a Valid Large-Sample Hypothesis Test for $σ^{2}$ $σ^{2}$

A random sample is selected from the target population.
The population from which the sample is selected has a distribution that is approximately normal.

FILLWTExample 6.13 A Two-Tailed Test for $σ^{2}$ $σ^{2}$ —Fill Weight Variance

Problem

Refer to Example 6.12. Now conduct the two-tailed test of hypothesis, $H_{0} : σ^{2} = .01$ $H_{0} : σ^{2} = .01$ against $H_{a} : σ^{2} \neq .01$ $H_{a} : σ^{2} \neq .01$ , using $α = .05$ $α = .05$ .

Solution

Since $α = .05$ $α = .05$ , $α / 2 = .025$ $α / 2 = .025$ and $(1 - α / 2) = .975$ $(1 - α / 2) = .975$ . The rejection region for this two-tailed test, as outlined in the box, requires that we find the critical chi-square values $χ_{(1 - α / 2)}^{2} = χ_{.975}^{2}$ $χ_{(1 - α / 2)}^{2} = χ_{.975}^{2}$ and $χ_{α / 2}^{2} = χ_{.025}^{2}$ $χ_{α / 2}^{2} = χ_{.025}^{2}$ . Looking in the 9 df row of Table IV Appendix B, we find $χ_{.975}^{2} = 2.70039$ $χ_{.975}^{2} = 2.70039$ and $χ_{.025}^{2} = 19.0228$ $χ_{.025}^{2} = 19.0228$ . Therefore, the rejection region is:

$Reject H_{0} if χ^{2} < 2.70039 or χ^{2} > 19.0228$ $Reject H_{0} if χ^{2} < 2.70039 or χ^{2} > 19.0228$

From Example 6.12, the test statistic is $χ^{2} = 1.53$ $χ^{2} = 1.53$ . Since this value falls into the rejection region, there is sufficient evidence to reject H₀ in favor of H_a. That is, there is evidence to indicate that the variance in the fill weights, $σ^{2}$ $σ^{2}$ , differs from the hypothesized value of .01.

Look Back

Remember to divide the value of $α$ $α$ in half to find the critical chi-square values in the rejection region for a two-tailed test of a population variance. Of course, you can always resort to statistical software to find the p-value of the test and derive your conclusion by simply comparing $α$ $α$ to the p-value.

Caution

The procedure for conducting a hypothesis test for $σ^{2}$ $σ^{2}$ in the preceding examples requires an assumption regardless of whether the sample size n is large or small. We must assume that the population from which the sample is selected has an approximate normal distribution. Unlike small-sample tests for $μ$ $μ$ based on the t-statistic, slight to moderate departures from normality will render the $χ^{2}$ $χ^{2}$ test invalid.

Exercises 6.94–6.114

Understanding the Principles

6.94 What sampling distribution is used to make inferences about $σ^{2} ?$ $σ^{2} ?$
6.95 What conditions are required for a valid test for $σ^{2} ?$ $σ^{2} ?$
6.96 True or False. The null hypotheses $H_{0} : σ^{2} = .25$ $H_{0} : σ^{2} = .25$ and $H_{0} : σ = .5$ $H_{0} : σ = .5$ are equivalent.
6.97 True or False. When the sample size n is large, no assumptions about the population are necessary to test the population variance $σ^{2} .$ $σ^{2} .$

Learning the Mechanics

6.98 Let $χ_{0}^{2}$ $χ_{0}^{2}$ be a particular value of $χ^{2} .$ $χ^{2} .$ Find the value of $χ_{0}^{2}$ $χ_{0}^{2}$ such that
1. $P (χ^{2} > χ_{0}^{2}) = .10$ $P (χ^{2} > χ_{0}^{2}) = .10$ for $n = 12$ $n = 12$
2. $P (χ^{2} > χ_{0}^{2}) = .05$ $P (χ^{2} > χ_{0}^{2}) = .05$ for $n = 9$ $n = 9$
3. $P (χ^{2} > χ_{0}^{2}) = .025$ $P (χ^{2} > χ_{0}^{2}) = .025$ for $n = 5$ $n = 5$
4. $P (χ^{2} < χ_{0}^{2}) = .05$ $P (χ^{2} < χ_{0}^{2}) = .05$ for $n = 7$ $n = 7$
5. $P (χ^{2} < χ_{0}^{2}) = .025$ $P (χ^{2} < χ_{0}^{2}) = .025$ for $n = 10$ $n = 10$
6.99 A random sample of n observations is selected from a normal population to test the null hypothesis that $σ^{2} = 25.$ $σ^{2} = 25.$ Specify the rejection region for each of the following combinations of $H_{a}, α,$ $H_{a}, α,$ and n:
1. $H_{a} : σ^{2} \neq 25; α = .05; n = 16$ $H_{a} : σ^{2} \neq 25; α = .05; n = 16$
2. $H_{a} : σ^{2} > 25; α = .01; n = 23$ $H_{a} : σ^{2} > 25; α = .01; n = 23$
3. $H_{a} : σ^{2} > 25; α = .10; n = 15$ $H_{a} : σ^{2} > 25; α = .10; n = 15$
4. $H_{a} : σ^{2} < 25; α = .01; n = 13$ $H_{a} : σ^{2} < 25; α = .01; n = 13$
5. $H_{a} : σ^{2} \neq 25; α = .10; n = 7$ $H_{a} : σ^{2} \neq 25; α = .10; n = 7$
6. $H_{a} : σ^{2} < 25; α = .05; n = 25$ $H_{a} : σ^{2} < 25; α = .05; n = 25$
6.100 A random sample of seven measurements gave $\overline{x} = 9.4$ $\overline{x} = 9.4$ and $s^{2} = 1.84.$ $s^{2} = 1.84.$
1. What assumptions must you make concerning the population in order to test a hypothesis about $σ^{2} ?$ $σ^{2} ?$
2. Suppose the assumptions in part a are satisfied. Test the null hypothesis $σ^{2} = 1$ $σ^{2} = 1$ against the alternative hypothesis $σ^{2} > 1.$ $σ^{2} > 1.$ Use $α = .05 .$ $α = .05 .$
3. Refer to part b. Suppose the test statistic is $χ^{2} = 14.45$ $χ^{2} = 14.45$ . Use Table IV of Appendix B or statistical software to find the p-value of the test.
4. Test the null hypothesis $σ^{2} = 1$ $σ^{2} = 1$ against the alternative hypothesis $σ^{2} \neq 1.$ $σ^{2} \neq 1.$ Use $α = .05 .$ $α = .05 .$
6.101 Refer to Exercise 6.100. Suppose we had $n = 100, \overline{x} = 9.4,$ $n = 100, \overline{x} = 9.4,$ and $s^{2} = 1.84.$ $s^{2} = 1.84.$
1. Test the null hypothesis $H_{0} : σ^{2} = 1$ $H_{0} : σ^{2} = 1$ against the alternative hypothesis $H_{a} : σ^{2} > 1.$ $H_{a} : σ^{2} > 1.$
2. Compare your test result with that of Exercise 6.114.
6.102 A random sample of $n = 7$ $n = 7$ observations from a normal population produced the following measurements: 4, 0, 6, 3, 3, 5, 9. Do the data provide sufficient evidence to indicate that $σ^{2} < 2 ?$ $σ^{2} < 2 ?$ Test, using $α = .05 .$ $α = .05 .$

Applying the Concepts—Basic

6.103 Trading skills of institutional investors. The Journal of Finance (Apr. 2011) published the results of an analysis of trading skills of institutional investors. The study focused on “round-trip” trades, i.e., trades in which the same stock was both bought and sold in the same quarter. In a random sample of 200 round-trip trades made by institutional investors, the sample standard deviation of the rates of return was 8.82%. One property of a consistent performance of institutional investors is a small variance in the rates of return of round-trip trades, say, a standard deviation of less than 10%.
1. Specify the null and alternative hypotheses for determining whether the population of institutional investors performs consistently.
2. Find the rejection region for the test using $α = .05 .$ $α = .05 .$
3. Interpret the value of $α$ $α$ in the words of the problem.
4. A MINITAB printout of the analysis is shown on pg. 349. Locate the test statistic and p-value on the printout.
5. Give the appropriate conclusion in the words of the problem.
6. What assumptions about the data are required for the inference to be valid?
ISR 6.104 Irrelevant speech effects. Refer to the Acoustical Science & Technology (Vol. 35, 2014) study of irrelevant speech effects, Exercise 6.47 (p. 330). Recall that 71 subjects performed a memorization task under two conditions: (1) with irrelevant background speech and (2) in silence. Descriptive statistics on the relative difference in the error rates (RDER) for the two conditions yielded $\overline{x} = 78.2$ $\overline{x} = 78.2$ percent and $s = 63.2$ $s = 63.2$ percent. Suppose the researchers theorize that the true standard deviation of the RDER values differs from 60 percent.
1. Set up the null and alternative hypotheses for testing the researchers’ theory.
2. Compute the test statistic.
3. Find the rejection region of the test using $α = .01$ $α = .01$ .
4. Use statistical software to find the p-value of the test.
5. Make the appropriate practical conclusion. State your answer in the words of the problem.
6. What assumption about the population of RDER values must hold true for the inference, part e, to be valid?
ROCKS 6.105 Characteristics of a rockfall. Refer toConsider the Environmental Geology (Vol. 58, 2009) simulation study of how far a block from a collapsing rockwall will bounce down a soil slope, Exercise 2.61 (p. 61). Rebound lengths (in meters) were estimated for 13 rock bounces. The data are repeated shown in the table. Descriptive statistics for the rebound lengths are shown on the SAS printout at the bottom of the page. Consider a test of hypothesis for the variation in rebound lengths for the theoretical population of rock bounces from the collapsing rockwall. In particular, a geologist wants to determine if the variance differs from 10 m².

Alternate View

10.94 13.71 11.38 7.26 17.83 11.92 11.87 5.44 13.35

4.90 5.85 5.10 6.77

Based on Paronuzzi, P. “Rockfall-induced block propagation on a soil slope, northern Italy.” Environmental Geology, Vol. 58, 2009 (Table 2).
1. Define the parameter of interest.
2. Specify the null and alternative hypotheses.
3. Compute the value of the test statistic.
4. Determine the rejection region for the test using $α = .10$ $α = .10$ .
5. Find the approximate p-value of the test.
6. Make the appropriate conclusion.
7. What condition must be satisfied in order for the inference, part f, to be valid?
SHAFTS 6.106 Shaft graves in ancient Greece. Refer toConsider the American Journal of Archaeology (Jan. 2014) study of sword shaft graves in ancient Greece, Exercise 2.37 (p. 50). The number of sword shafts buried at each of 13 recently discovered grave sites is reproduced in the following table. Let $σ$ $σ$ represent the standard deviation of the number of sword shafts for the population of all shaft graves in ancient Greece. Consider the test $H_{0} : σ = 3$ $H_{0} : σ = 3$ versus $H_{a} : σ < 3$ $H_{a} : σ < 3$ , using $α = .05$ $α = .05$ .

Alternate View

1 2 3 1 5 6 2 4 1 2 4 2 9

Source: Harrell, K. “The fallen and their swords: A new explanation for the rise of the shaft graves.” American Journal of Archaeology, Vol. 118, No. 1, Jan. 2014 (Figure 1).
1. Use statistical software to find the sample standard deviation, s.
2. Calculate the test statistic.
3. Use statistical software to find the p-value of the test. Interpret the result.
4. Assume the sample grave with 9 sword shafts was discovered to be an outlying observation by the researchers, i.e., one from a different population. Remove the data point from the sample and repeat parts a–c. What conclusions can you now draw?
SAS Output for Exercise 6.105
6.107 Latex allergy in health care workers. Refer toConsider the Current Allergy & Clinical Immunology (Mar. 2004) study of $n = 46$ $n = 46$ hospital employees who were diagnosed with a latex allergy from exposure to the powder on latex gloves, presented in Exercise 5.15 (p. 262). Recall that theThe number of latex gloves used per week by the sampled workers is summarized as follows: $\overline{x} = 19.3$ $\overline{x} = 19.3$ and $s = 11.9.$ $s = 11.9.$ Let $σ^{2}$ $σ^{2}$ represent the variance in the number of latex gloves used per week by all hospital employees. Consider testing $H_{0} : σ^{2} = 100$ $H_{0} : σ^{2} = 100$ against $H_{a} : σ^{2} \neq 100.$ $H_{a} : σ^{2} \neq 100.$
1. Give the rejection region for the test at a significance level of $α = .01 .$ $α = .01 .$
2. Calculate the value of the test statistic.
3. Use the results from parts a and b to draw the appropriate conclusion.

Applying the Concepts—Intermediate

6.108 Oil content of fried sweet potato chips. Refer toConsider the Journal of Food Engineering (Sept. 2013) study of the characteristics of sweet potato chips fried at different temperatures, Exercise 5.111 (p. 294). Recall that aA sample of 6 sweet potato slices were fried at 130° using a vacuum fryer and the internal oil content (gigagrams) was measured for each slice. The results were: $\bar{x} = .178 g/g$ $\bar{x} = .178 g/g$ and $s = .011 g/g$ $s = .011 g/g$ .
1. Conduct a test of hypothesis to determine if the standard deviation, $σ$ $σ$ , of the population of internal oil contents for sweet potato slices fried at 130° differs from .1. Use $α = .05$ $α = .05$ .
2. In Exercise 5.111, you formedUse a 95% confidence interval for the true standard deviation of the internal oil content distribution for the sweet potato chips. Use this interval to make an inference about whether $σ = .1$ $σ = .1$ . Does the result agree with the test, part a?
FORCE 6.109 Strand bond performance of pre-stressed concrete. An experiment was carried out to investigate the strength of pre-stressed, bonded concrete after anchorage failure has occurred and the results published in Engineering Structures (June 2013). The maximum strand force, measured in kiloNewtons (kN), achieved after anchorage failure for 12 pre-stressed concrete strands is given in the accompanying table. Conduct a test of hypothesis to determine if the true standard deviation of the population of maximum strand forces is less than 5 kN. Test using $α = .10$ $α = .10$ .

Alternate View

158.2 161.5 166.5 158.4 159.9 161.9 162.8 161.2 160.1 175.6 168.8 163.7

TABLET 6.110 Drug content assessment. Analytical Chemistry (Dec. 15, 2009) presented research on a new method used by GlaxoSmithKline Medicines Research Center to determine the amount of drug in a tablet. Drug concentrations (measured as a percentage) for 50 randomly selected tablets are provided in the accompanying table. The standard method of assessing drug content yields a concentration variance of 9. Can the scientists at GlaxoSmithKline conclude that the new method of determining drug concentration is less variable than the standard method? Test using $α = .01$ $α = .01$ .

Alternate View

91.28 92.83 89.35 91.90 82.85 94.83 89.83 89.00 84.62

86.96 88.32 91.17 83.86 89.74 92.24 92.59 84.21 89.36

90.96 92.85 89.39 89.82 89.91 92.16 88.67 89.35 86.51

89.04 91.82 93.02 88.32 88.76 89.26 90.36 87.16 91.74

86.12 92.10 83.33 87.61 88.20 92.78 86.35 93.84 91.20

93.44 86.77 83.77 93.19 81.79

Source: Borman, P. J., Marion, J. C., Damjanov, I., & Jackson, P. “Design and analysis of method equivalence studies.” Analytical Chemistry, Vol. 81, No. 24, Dec. 15, 2009 (Table 3).

6.111 Point spreads of NFL games. During the National Football League (NFL) season, Las Vegas oddsmakers establish a point spread on each game for betting purposes. For example, the defending champion Seattle Seahawks were established as a 1-point underdog against the New England Patriots in the 2015 Super Bowl. The final scores of NFL games were compared against the final point spreads established by the oddsmakers in Chance (Fall 1998). The difference between the outcome of the game and the point spread (called a point-spread error) was calculated for 240 NFL games. The mean and standard deviation of the point-spread errors are $\overline{x} = - 1.6$ $\overline{x} = - 1.6$ and $s = 13.3.$ $s = 13.3.$ Suppose the researcher wants to know whether the true standard deviation of the point-spread errors exceeds 15. Conduct the analysis using $α = .05 .$ $α = .05 .$
TURBINE 6.112 Cooling method for gas turbines. Refer to the Journal of Engineering for Gas Turbines and Power (Jan. 2005) study of the performance of augmented gas turbine engines, presented in Exercise 6.46 (p. 330). Recall that the performance of each in a sample of 67 gas turbines was measured by heat rate (kilojoules per kilowatt per hour). Suppose that standard gas turbines have heat rates with a standard deviation of 1,500 kJ $/$ $/$ kWh. Is there sufficient evidence to indicate that the heat rates of the augmented gas turbine engine are more variable than the heat rates of the standard gas turbine engine? Test, using $α = 0.5.$ $α = 0.5.$
6.113 Birth weights of cocaine babies. A group of researchers at the University of Texas-Houston conducted a comprehensive study of pregnant cocaine-dependent women (Journal of Drug Issues, Summer 1997). All the women in the study used cocaine on a regular basis (at least three times a week) for more than a year. One of the many variables measured was birth weight (in grams) of the baby delivered. For a sample of 16 cocaine-dependent women, the mean birth weight was 2,971 grams and the standard deviation was 410 grams. Test (at $α = .01$ $α = .01$ ) to determine whether the variance in birth weights of babies delivered by cocaine-dependent women is less than $200,000 {grams}^{2} .$ $200,000 {grams}^{2} .$

Applying the Concepts—Advanced

6.114 Motivation of drug dealers. Refer toConsider the Applied Psychology in Criminal Justice (Sept. 2009) study of the personality characteristics of convicted drug dealers, Exercise 5.17 (p. 262). A random sample of 100 drug dealers had a mean Wanting Recognition (WR) score of 39 points, with a standard deviation of 6 points. Recall that theThe WR score is a quantitative measure of a person’s level of need for approval and sensitivity to social situations. (Higher scores indicate a greater need for approval.) A criminal psychologist claims that the range of WR scores for the population of convicted drug dealers is 42 points. Do you believe the psychologist’s claim? (Hint: Assume the population of WR scores is normally distributed.)

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Table of Contents for
6.7 Test of Hypothesis about a Population Variance (Optional)

6.7 Test of Hypothesis about a Population Variance (Optional)

Table 6.6 Fill Weights (ounces) of 10 Cans

Biography Friedrich R. Helmert (1843–1917)

FILLWT Example 6.12 A One-Tailed Test for $σ^{2}$ $σ^{2}$ —Fill Weight Variance

Problem

Solution

Figure 6.24 Rejection region for Example 6.12

Figure 6.25 SAS test of fill amount variance, Example 6.12

Look Back

Large-Sample Test of Hypothesis about $σ^{2 :}$ $σ^{2 :}$ Chi-Square Statistic

Conditions Required for a Valid Large-Sample Hypothesis Test for $σ^{2}$ $σ^{2}$

FILLWTExample 6.13 A Two-Tailed Test for $σ^{2}$ $σ^{2}$ —Fill Weight Variance

Problem

Solution

Look Back

Caution

Exercises 6.94–6.114

Understanding the Principles

Learning the Mechanics

Applying the Concepts—Basic

MINITAB Output for Exercise 6.103

SAS Output for Exercise 6.105

Applying the Concepts—Intermediate

Applying the Concepts—Advanced

16.00	15.95	16.10	16.02	15.99
16.06	16.04	16.05	16.03	16.02

10.94	13.71	11.38	7.26	17.83	11.92	11.87	5.44	13.35
4.90	5.85	5.10	6.77

91.28	92.83	89.35	91.90	82.85	94.83	89.83	89.00	84.62
86.96	88.32	91.17	83.86	89.74	92.24	92.59	84.21	89.36
90.96	92.85	89.39	89.82	89.91	92.16	88.67	89.35	86.51
89.04	91.82	93.02	88.32	88.76	89.26	90.36	87.16	91.74
86.12	92.10	83.33	87.61	88.20	92.78	86.35	93.84	91.20
93.44	86.77	83.77	93.19	81.79

Table of Contents for 6.7 Test of Hypothesis about a Population Variance (Optional)

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Table of Contents for
6.7 Test of Hypothesis about a Population Variance (Optional)