Step 2 The confidence interval for σ is from
11.79 A Better Golf Tee? Refer to Exercise 11.73, and obtain
11.80 Nitrogen and Seagrass. Refer to Exercise 11.74, and ob- tain a 98% confidence interval for the ratio of the population stan- dard deviations of sediment ammonium concentrations for LLM seagrass beds and CCB seagrass beds. (Note:For df=(18,50), F0.01=2.32.)
Working with Large Data Sets
11.81 The Etruscans.Anthropologists are still trying to unravel the mystery of the origins of the Etruscan empire, a highly ad- vanced Italic civilization formed around the eighth century B.C.
in central Italy. Were they native to the Italian peninsula or, as many aspects of their civilization suggest, did they migrate from
540 CHAPTER 11 Inferences for Population Standard Deviations∗ the East by land or sea? The maximum head breadth, in millime- ters, of 70 modern Italian male skulls and that of 84 preserved Etruscan male skulls were analyzed to help researchers decide whether the Etruscans were native to Italy. The resulting data can be found on the WeissStats CD. [SOURCE: N. Barnicot and D. Brothwell, “The Evaluation of Metrical Data in the Compari- son of Ancient and Modern Bones.” InMedical Biology and Etr- uscan Origins, G. E. W. Wolstenholme and C. M. O’Connor, eds., Little, Brown & Co., 1959] Use the technology of your choice to solve parts (a)–(c).
a. Perform a two-standard-deviations F-test at the 5% signifi- cance level to decide whether the data provide sufficient ev- idence to conclude that variation in skull measurements differ between the two populations.
b. Use the two-standard-deviationsF-interval procedure to de- termine a 95% confidence interval for the ratio of the standard deviations of skull measurements of the two populations.
c. Obtain a normal probability plot for each sample.
d. In light of your plots in part (c), does conducting the infer- ences you did in parts (a) and (b) seem reasonable? Explain your answer.
11.82 Active Management of Labor. Active management of labor (AML) is a group of interventions designed to help reduce the length of labor and the rate of cesarean deliveries. Physicians from the Department of Obstetrics and Gynecology at theUni- versity of New Mexico Health Sciences Centerwere interested in determining whether AML would affect the cost of delivery.
The results of their study can be found in Rogers et al., “Ac- tive Management of Labor: A Cost Analysis of a Randomized Controlled Trial” (Western Journal of Medicine, Vol. 172, pp. 240–243). Data based on the researchers’ findings on the cost of cesarean deliveries for independent random samples of those using AML and those using standard hospital protocols are pro- vided on the WeissStats CD. Use the technology of your choice to solve parts (a)–(c).
a. Do the data provide sufficient evidence to conclude that the variation in cost is greater with AML than without? Perform a two-standard-deviationsF-test at the 10% significance level.
b. Use the two-standard-deviationsF-interval procedure to deter- mine an 80% confidence interval for the ratio of the population standard deviations of costs with and without AML.
c. Obtain a normal probability plot for each sample.
d. In light of your plots in part (c), does conducting the infer- ences you did in parts (a) and (b) seem reasonable? Explain your answer.
11.83 RBC Transfusions. In the article “Reduction in Red Blood Cells Transfusions Among Preterm Infants: Results of a Randomized Trial With an In-Line Blood Gas and Chem- istry Monitor” (Pediatrics, Vol. 115, Issue 5, pp. 1299–1306), J. Widness et al. examined extremely premature infants who
develop anemia caused by intensive laboratory blood testing and multiple red blood cell (RBC) transfusions. The goal of the study was to reduce the number of RBC transfusions. Two groups were studied, a control group and a monitor group (which used the in- line blood gas and chemistry monitor). Data on hemoglobin level, in grams per liter (g/L), based on the results of the study, are pro- vided on the WeissStats CD. Use the technology of your choice to solve parts (a)–(c).
a. Do the data provide sufficient evidence to conclude that the variation in hemoglobin level is less without the inline blood gas and chemistry monitor? Perform a two-standard- deviationsF-test at the 5% significance level.
b. Use the two-standard-deviationsF-interval procedure to deter- mine a 90% confidence interval for the ratio of the population standard deviations of hemoglobin levels with and without the inline blood gas and chemistry monitor.
c. Obtain a normal probability plot for each sample.
d. In light of your plots in part (c), does conducting the infer- ences you did in parts (a) and (b) seem reasonable? Explain your answer.
Extending the Concepts and Skills
11.84 Simulation. Use the technology of your choice to con- duct the simulation discussed in Example 11.13 on page 530.
11.85 Elmendorf Tear Strength. Refer to Example 11.14 on page 530. Use Table VIII to show that the P-value for the hy- pothesis test exceeds 0.20.
11.86 Because of space restrictions, the numbers of degrees of freedom in Table VIII are not consecutive. For instance, the de- grees of freedom for the numerator skips from 24 to 30. If you had only Table VIII and you needed to find F0.05for df=(25,20), how would you do it?
EstimatingF-values From Table VIII.One solution to Exer- cise 11.86 is to use linear interpolation as follows: For df = (24, 20), we haveF0.05=2.08; and for df=(30, 20), we have F0.05=2.04. Because 25 is 1/6 of the way between 24 and 30, we estimate that for anF-curve with df=(25, 20),
F0.05=2.08+1
6ã(2.04−2.08)=2.07.
In Exercises11.87–11.90, use Table VIII and linear interpolation to estimate the required F-values.
11.87 F0.10for df=(25,15).
11.88 F0.05for df=(8,90). 11.89 F0.05for df=(19,40).
11.90 F0.01for df=(18,50).
CHAPTER IN REVIEW
You Should Be Able to
1. use and understand the formulas in this chapter.
2. state the basic properties ofχ2-curves.
3. use the chi-square table, Table VII.
4. perform a hypothesis test for a population standard devia- tion when the variable under consideration is normally dis- tributed.
5. obtain a confidence interval for a population standard de- viation when the variable under consideration is normally distributed.
6. state the basic properties ofF-curves.
7. apply the reciprocal property ofF-curves.
8. use theF-table, Table VIII.
9. perform a hypothesis test to compare two population stan- dard deviations when the variable under consideration is nor- mally distributed on both populations.
10. find a confidence interval for the ratio of two population stan- dard deviations when the variable under consideration is nor- mally distributed on both populations.
Key Terms
χα2,513
chi-square(χ2)curve,512 chi-square distribution,512 degrees of freedom for the
denominator,526
degrees of freedom for the numerator,526 Fα,527
F-curve,526 F-distribution,526 F-statistic,529
one-standard-deviationχ2-interval procedure,519
one-standard-deviationχ2-test,517 two-standard-deviationsF-interval
procedure,533
two-standard-deviationsF-test,531
REVIEW PROBLEMS
Understanding the Concepts and Skills
1. What distribution is used in this chapter to make inferences for one population standard deviation?
2. Fill in the blanks.
a. Aχ2-curve is skewed.
b. Aχ2-curve looks increasingly like a curve as the num- ber of degrees of freedom becomes larger.
3. When you use the one-standard-deviation χ2-test or χ2-interval procedure, what assumption must be met by the vari- able under consideration? How important is that assumption?
4. Consider a χ2-curve with 17 degrees of freedom. Use Table VII to determine
a. χ0.992 . b. χ0.012 . c. theχ2-value having area 0.05 to its right.
d. theχ2-value having area 0.05 to its left.
e. the twoχ2-values that divide the area under the curve into a middle 0.95 area and two outside 0.025 areas.
5. What distribution is used in this chapter to make inferences for two population standard deviations?
6. Fill in the blanks:
a. AnF-curve is skewed.
b. For an F-curve with df = (14,5), the F-value having area 0.05 to its left equals the of the F-value having area 0.05 to its right for an F-curve with df =
( , ).
c. The observed value of a variable having an F-distribution must be greater than or equal to .
7. When you use the two-standard-deviations F-test, what as- sumption must be met by the variable under consideration? How important is that assumption?
8. Consider an F-curve with df = (4, 8). Use Table VIII to determine
a. F0.01. b. F0.99.
c. theF-value having area 0.05 to its right.
d. theF-value having area 0.05 to its left.
e. the two F-values that divide the area under the curve into a middle 0.95 area and two outside 0.025 areas.
9. Intelligence Quotients. IQs measured on the Stanford Re- vision of the Binet–Simon Intelligence Scale are supposed to have a standard deviation of 16 points. Twenty-five randomly se- lected people were given the IQ test; here are the data that were obtained.
91 96 106 116 97
102 96 124 115 121
95 111 105 101 86
88 129 112 82 98
104 118 127 66 102
Preliminary data analyses and other information indicate the reasonableness of presuming that IQs measured on the Stanford Revision of the Binet–Simon Intelligence Scale are normally dis- tributed.
a. Do the data provide sufficient evidence to conclude that IQs measured on this scale have a standard deviation different from 16 points? Perform the required hypothesis test at the 10% significance level. (Note: s=15.006.)
b. How crucial is the normality assumption for the hypothesis test you performed in part (a)? Explain your answer.
10. Intelligence Quotients. Refer to Problem 9. Determine a 90% confidence interval for the standard deviation of IQs mea- sured on the Stanford Revision of the Binet–Simon Intelligence Scale.
11. Skinfold Thickness. A study entitled “Body Composition of Elite Class Distance Runners” was conducted by M. Pollock et al. to decide whether elite distance runners are thinner than other people. Their results were published in The Marathon:
542 CHAPTER 11 Inferences for Population Standard Deviations∗ Physiological, Medical, Epidemiological, and Psychological Studies, P. Milvey (ed.), New York: New York Academy of Sci- ences, 1977, p. 366. The researchers measured the skinfold thick- ness, an indirect indicator of body fat, of runners and nonrunners in the same age group. The data, in millimeters (mm), shown in the following table are based on the skinfold thickness measure- ments on the thighs of the people sampled.
Runners Others
7.3 6.7 8.7 24.0 19.9 7.5 18.4 3.0 5.1 8.8 28.0 29.4 20.3 19.0 7.8 3.8 6.2 9.3 18.1 22.8 24.2 5.4 6.4 6.3 9.6 19.4 16.3 16.3 3.7 7.5 4.6 12.4 5.2 12.2 15.6
a. For anF-test to compare the standard deviations of skinfold thickness of runners and others, identify the appropriate F- distribution.
b. At the 1% significance level, do the data provide sufficient evi- dence to conclude that runners have less variability in skinfold thickness than others? (Note: s1=1.798 ands2=6.606. For df=(19,14),F0.01=3.53.)
c. What assumption about skinfold thickness did you make in carrying out the hypothesis test in part (b)? How would you check that assumption?
d. In addition to the assumption on skinfold thickness discussed in part (c), what other assumptions are required for perform- ing the two-standard-deviationsF-test?
12. Skinfold Thickness. Refer to Problem 11. Find a 98% con- fidence interval for the ratio of the standard deviations of skinfold thickness for runners and for others. (Note:For df= (14,19), F0.01=3.19.)
Working with Large Data Sets
13. Body Mass Index. Body mass index (BMI) is a measure of body fat based on height and weight. According to the document Dietary Guidelines for Americans published by the U.S. De- partment of Agricultureand theU.S. Department of Health and Human Services, for adults, a BMI of greater than 25 indicates an above healthy weight (i.e., overweight or obese). The BMIs of 75 randomly selected U.S. adults provided the data on the WeissStats CD. Use the technology of your choice to do the following.
a. Obtain a normal probability plot, a boxplot, and a histogram of the data.
b. Based on your graphs from part (a), is it reasonable to ap- ply one-standard-deviation χ2-procedures to the data? Ex- plain your answer.
c. In Problem 40 of Chapter 9, we applied the one-meanz-test to the data, assuming a standard deviation of 5.0 for the BMIs of all U.S. adults. At the 5% significance level, do the data provide evidence against that assumption?
14. Body Mass Index. Refer to Problem 13, and find a 95% con- fidence interval for the standard deviation of BMIs for all U.S. adults.
15. Gender and Direction. In the paper “The Relation of Sex and Sense of Direction to Spatial Orientation in an Unfamiliar Environment” (Journal of Environmental Psychology, Vol. 20, pp. 17–28), J. Sholl et al. published the results of examining the sense of direction of 30 male and 30 female students. After being taken to an unfamiliar wooded park, the students were given a number of spatial orientation tests, including pointing to south, which tested their absolute frame of reference. To point south, the students moved a pointer attached to a 360◦protractor.
The absolute pointing errors, in degrees, for students who rated themselves with a good sense of direction (GSOD) and those who rated themselves with a poor sense of direction (PSOD) are pro- vided on the WeissStats CD. Can you reasonably apply the two- standard-deviations F-test to compare the variation in pointing errors between people who rate themselves with a good sense of direction and those who rate themselves with a poor sense of di- rection? Explain your answer.
16. Microwave Popcorn. Two brands of microwave popcorn, which we will call Brand A and Brand B, were compared for consistency in popping time. The popping times, in seconds, for 30 bags of each brand are provided on the WeissStats CD. Use the technology of your choice to do the following.
a. Obtain normal probability plots and boxplots, and histograms for the two data sets.
b. Based on your graphs from part (a), do you think it reason- able to perform a two-standard-deviationsF-test on the data?
Explain your answer.
c. At the 5% significance level, do the data provide sufficient evi- dence to conclude that Brand B has a more consistent popping time than Brand A?
d. Find a 90% confidence interval for the ratio of the standard deviations of popping times for Brand A and Brand B.
FOCUSING ON DATA ANALYSIS
UWEC UNDERGRADUATES Recall from Chapter 1 (see pages 30–31) that the Focus database and Focus sample contain information on the un- dergraduate students at the University of Wisconsin - Eau Claire (UWEC). Now would be a good time for you to re- view the discussion about these data sets.
Open the Focus sample worksheet (FocusSample) in the technology of your choice and then do the following.
a. At the 5% significance level, do the data provide suf- ficient evidence to conclude that the standard deviation of ACT composite scores of all UWEC undergraduates differs from 3 points?
b. Determine and interpret a 95% confidence interval for the standard deviation of ACT composite scores of all UWEC undergraduates.
c. Obtain a normal probability plot and a boxplot of the ACT composite scores of the sampled UWEC undergraduates.
d. Based on your results from part (c), do you think that performing the inferences in parts (a) and (b) is reason- able? Explain your answer.
e. At the 5% significance level, do the data provide suffi- cient evidence to conclude that the standard deviations of ACT English scores and ACT math scores differ for UWEC undergraduates?
f. Determine and interpret a 95% confidence interval for the ratio of the standard deviation of ACT English scores to the standard deviation of ACT math scores for UWEC undergraduates.
g. Obtain normal probability plots and boxplots of the ACT English scores and the ACT math scores of the sampled UWEC undergraduates.
h. Based on your results from part (g), do you think that performing the inference in parts (e) and (f ) is reason- able? Explain your answer.
CASE STUDY DISCUSSION