In the paper “Schizophrenia: Dopamineβ-Hydroxylase Activity and Treatment Response” (Science, Vol. 216, pp. 1423–1425), D. Sternberg et al. published the results of their study in which they examined 25 schizophrenic patients who had been classified as either psychotic or not psychotic by hospital staff. The activ- ity of dopamine was measured in each patient by using the en- zyme dopamineβ-hydroxylase to assess differences in dopamine activity between the two groups. The following are the data, in nanomoles per milliliter-hour per milligram (nmol/mL-hr/mg).
Psychotic Not psychotic 0.0150 0.0222 0.0104 0.0230 0.0145 0.0204 0.0275 0.0200 0.0116 0.0180 0.0306 0.0270 0.0210 0.0252 0.0154 0.0320 0.0226 0.0105 0.0130 0.0170 0.0208 0.0245 0.0112 0.0200 0.0156
At the 1% significance level, do the data suggest that dopamine activity is higher, on average, in psychotic patients?
(Note: x¯1=0.02426, s1=0.00514, x¯2=0.01643, and s2= 0.00470.)
10.74 Wing Length. D. Cristol et al. published results of their studies of two subspecies of dark-eyed juncos in the article “Mi- gratory Dark-Eyed Juncos,Junco Hyemalis, Have Better Spatial Memory and Denser Hippocampal Neurons than Nonmigratory Conspecifics” (Animal Behaviour, Vol. 66, pp. 317–328). One of the subspecies migrates each year, and the other does not mi- grate. Several physical characteristics of 14 birds of each sub- species were measured, one of which was wing length. The fol- lowing data, based on results obtained by the researchers, provide the wing lengths, in millimeters (mm), for the samples of two subspecies.
Migratory Nonmigratory 84.5 81.0 82.6 82.1 82.4 83.9 82.8 84.5 81.2 87.1 84.6 85.1 80.5 82.1 82.3 86.3 86.6 83.9 80.1 83.4 81.7 84.2 84.3 86.2
83.0 79.7 87.8 84.1
a. At the 1% significance level, do the data provide sufficient evidence to conclude that the mean wing lengths for the two subspecies are different? (Note: The mean and stan- dard deviation for the migratory-bird data are 82.1 mm and 1.501 mm, respectively, and that for the nonmigratory- bird data are 84.9 mm and 1.698 mm, respectively.)
b. Would it be reasonable to use a pooledt-test here? Explain your answer.
c. If your answer to part (b) wasyes, then perform a pooledt-test to answer the question in part (a) and compare your results to that found in part (a) by using a nonpooledt-test.
In Exercises10.75–10.80, apply Procedure 10.4 on page 456 to obtain the required confidence interval. Interpret your result in each case.
10.75 Political Prisoners. Refer to Exercise 10.69 and obtain a 90% confidence interval for the difference,μ1−μ2, between the mean ages at arrest of East German prisoners with chronic PTSD and remitted PTSD.
10.76 Nitrogen and Seagrass. Refer to Exercise 10.70 and de- termine a 98% confidence interval for the difference,μ1−μ2, between the mean sediment ammonium concentrations in CCB and LLM.
10.77 Acute Postoperative Days. Refer to Exercise 10.71 and find a 90% confidence interval for the difference between the mean numbers of acute postoperative days in the hospital with the dynamic and static systems.
10.78 Stressed-Out Bus Drivers. Refer to Exercise 10.72 and find a 90% confidence interval for the difference between the mean heart rates of urban bus drivers in Stockholm in the two environments.
10.79 Schizophrenia and Dopamine. Refer to Exercise 10.73 and determine a 98% confidence interval for the difference be- tween the mean dopamine activities of psychotic and nonpsy- chotic patients.
10.80 Wing Length. Refer to Exercise 10.74 and find a 99% confidence interval for the difference between the mean wing lengths of the two subspecies.
462 CHAPTER 10 Inferences for Two Population Means 10.81 Sleep Apnea. In the article “Sleep Apnea in Adults With Traumatic Brain Injury: A Preliminary Investigation” (Archives of Physical Medicine and Rehabilitation, Vol. 82, Issue 3, pp. 316–321), J. Webster et al. investigated sleep-related breath- ing disorders in adults with traumatic brain injuries (TBI). The respiratory disturbance index (RDI), which is the number of ap- neic and hypopneic episodes per hour of sleep, was used as a measure of severity of sleep apnea. An RDI of 5 or more indi- cates sleep-related breathing disturbances. The RDIs for the fe- males and males in the study are as follows.
Female Male
0.1 0.5 0.3 2.3 2.6 19.3 1.4 1.0 0.0 39.2 4.1 2.0 1.4 0.0 0.0 2.1 1.1 5.6 5.0 7.0 2.3 4.3 7.5 16.5 7.8 3.3 8.9 7.3 Use the technology of your choice to answer the following ques- tions. Explain your answers.
a. If you had to choose between the use of pooled t-procedures and nonpooledt-procedures here, which would you choose?
b. Is it reasonable to use the type of procedure that you selected in part (a)?
10.82 Mandate Perceptions. L. Grossback et al. examined mandate perceptions and their causes in the paper “Compar- ing Competing Theories on the Causes of Mandate Percep- tions” (American Journal of Political Science, Vol. 49, Issue 2, pp. 406–419). Following are data on the percentage of members in each chamber of Congress who reacted to mandates in various years.
House Senate
30.3 41.1 15.6 10.1 21 38 40 39 27
23.9 15.2 11.7 27 17 25 25
Use the technology of your choice to answer the following ques- tions. Explain your answers.
a. If you had to choose between the use of pooled t-procedures and nonpooledt-procedures here, which would you choose?
b. Is it reasonable to use the type of procedure that you selected in part (a)?
10.83 Acute Postoperative Days. In Exercise 10.71, you con- ducted a nonpooledt-test to decide whether the mean number of
acute postoperative days spent in the hospital is smaller with the dynamic system than with the static system.
a. Using a pooledt-test, repeat that hypothesis test.
b. Compare your results from the pooled and nonpooledt-tests.
c. Which test do you think is more appropriate, the pooled or nonpooledt-test? Explain your answer.
10.84 Neurosurgery Operative Times. In Example 10.6 on page 454, we conducted a nonpooled t-test, at the 5% signifi- cance level, to decide whether the mean operative time is less with the dynamic system than with the static system.
a. Using a pooledt-test, repeat that hypothesis test.
b. Compare your results from the pooled and nonpooledt-tests.
c. Repeat both tests, using a 1% significance level, and compare your results.
d. Which test do you think is more appropriate, the pooled or nonpooledt-test? Explain your answer.
10.85 Each pair of graphs in Fig. 10.8 shows the distributions of a variable on two populations. Suppose that, in each case, you want to perform a small-sample hypothesis test based on inde- pendent simple random samples to compare the means of the two populations. In each case, decide whether the pooledt-test, non- pooledt-test, or neither should be used. Explain your answers.
Working with Large Data Sets
10.86 Treating Psychotic Illness. L. Petersen et al. evaluated the effects of integrated treatment for patients with a first episode of psychotic illness in the paper “A Randomised Multicentre Trial of Integrated Versus Standard Treatment for Patients With a First Episode of Psychotic Illness” (British Medical Journal, Vol. 331, (7517):602). Part of the study included a question- naire that was designed to measure client satisfaction for both the integrated treatment and a standard treatment. The data on the WeissStats CD are based on the results of the client question- naire. Use the technology of your choice to do the following.
a. Obtain normal probability plots, boxplots, and the standard deviations for the two samples.
b. Based on your results from part (a), which would you be in- clined to use to compare the population means: a pooled or a nonpooledt-procedure? Explain your answer.
c. Do the data provide sufficient evidence to conclude that, on average, clients preferred the integrated treatment? Perform the required hypothesis test at the 1% significance level by us- ing both the pooledt-test and the nonpooledt-test. Compare your results.
d. Find a 98% confidence interval for the difference be- tween mean client satisfaction scores for the two treatments.
FIGURE 10.8 Figure for Exercise 10.85
(a) (b)
(c) (d)
Obtain the required confidence interval by using both the pooledt-interval procedure and the nonpooledt-interval pro- cedure. Compare yours results.
10.87 A Better Golf Tee? An independent golf equipment testing facility compared the difference in the performance of golf balls hit off a regular 2-3/4wooden tee to those hit off a 3 Stinger Competition golf tee. A Callaway Great Big Bertha driver with 10 degrees of loft was used for the test and a robot swung the club head at approximately 95 miles per hour. Data on ball velocity (in miles per hour) with each type of tee, based on the test results, are provided on the WeissStats CD. Use the technology of your choice to do the following.
a. Obtain normal probability plots, boxplots, and the standard deviations for the two samples.
b. Based on your results from part (a), which would you be in- clined to use to compare the population means: a pooled or a nonpooledt-procedure? Explain your answer.
c. At the 5% significance level, do the data provide sufficient ev- idence to conclude that, on average, ball velocity is less with the regular tee than with the Stinger tee? Perform the required hypothesis test by using both the pooledt-test and the non- pooledt-test, and compare results.
d. Find a 90% confidence interval for the difference between the mean ball velocities with the regular and Stinger tees. Ob- tain the required confidence interval by using both the pooled t-interval procedure and the nonpooled t-interval procedure.
Compare your results.
10.88 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 the East by land or sea? The maximum head breadth, in millimeters, of 70 modern Italian male skulls and 84 preserved Etruscan male skulls was 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. Wolstenholme and C. O’Connor, eds., Little, Brown & Co., 1959]
a. Obtain normal probability plots, boxplots, and the standard deviations for the two samples.
b. Based on your results from part (a), which would you be in- clined to use to compare the population means: a pooled or a nonpooledt-procedure? Explain your answer.
c. Do the data provide sufficient evidence to conclude that a dif- ference exists between the mean maximum head breadths of modern Italian males and Etruscan males? Perform the re- quired hypothesis test at the 5% significance level by using both the pooledt-test and the nonpooledt-test. Compare your results.
d. Find a 95% confidence interval for the difference between the mean maximum head breadths of modern Italian males and Etruscan males. Obtain the required confidence interval by us- ing both the pooled t-interval procedure and the nonpooled t-interval procedure. Compare your results.
Extending the Concepts and Skills
10.89 Suppose that the sample sizes, n1 andn2, are equal for independent simple random samples from two populations.
a. Show that the values of the pooled and nonpooledt-statistics will be identical. (Hint:Refer to Exercise 10.55 on page 451.) b. Explain why part (a) does not imply that the twot-tests are equivalent (i.e., will necessarily lead to the same conclusion) when the sample sizes are equal.
10.90 Tukey’s Quick Test. In this exercise, we examine an al- ternative method, conceived by the late Professor John Tukey, for performing a two-tailed hypothesis test for two population means based on independent random samples. To apply this pro- cedure, one of the samples must contain the largest observation (high group) and the other sample must contain the smallest ob- servation (low group). Here are the steps for performing Tukey’s quick test.
Step 1 Count the number of observations in the high group that are greater than or equal to the largest observation in the low group. Count ties as 1/2.
Step 2 Count the number of observations in the low group that are less than or equal to the smallest observation in the high group. Count ties as 1/2.
Step 3 Add the two counts obtained in Steps 1 and 2, and denote the sumc.
Step 4 Reject the null hypothesis at the 5% significance level if and only ifc≥7; reject it at the 1% significance level if and only ifc≥10; and reject it at the 0.1% significance level if and only ifc≥13.
a. Can Tukey’s quick test be applied to Exercise 10.42 on page 450? Explain your answer.
b. If your answer to part (a) wasyes, apply Tukey’s quick test and compare your result to that found in Exercise 10.42, where a t-test was used.
c. Can Tukey’s quick test be applied to Exercise 10.74? Explain your answer.
d. If your answer to part (c) wasyes, apply Tukey’s quick test and compare your result to that found in Exercise 10.74, where a t-test was used.
For more details about Tukey’s quick test, see J. Tukey, “A Quick, Compact, Two-Sample Test to Duckworth’s Specifica- tions” (Technometrics, Vol. 1, No. 1, pp. 31–48).
10.91 Two-Tailed Hypothesis Tests and CIs.As we mentioned on page 446, the following relationship holds between hypothe- sis tests and confidence intervals: For a two-tailed hypothesis test at the significance levelα, the null hypothesis H0:μ1=μ2will be rejected in favor of the alternative hypothesisHa:μ1 =μ2if and only if the (1−α)-level confidence interval forμ1−μ2does not contain 0. In each case, illustrate the preceding relationship by comparing the results of the hypothesis test and confidence interval in the specified exercises.
a. Exercises 10.69 and 10.75 b. Exercises 10.74 and 10.80 10.92 Left-Tailed Hypothesis Tests and CIs. If the as- sumptions for a nonpooledt-interval are satisfied, the formula for a (1−α)-level upper confidence bound for the difference, μ1−μ2, between two population means is
(x¯1− ¯x2)+tαã
(s12/n1)+(s22/n2).
For a left-tailed hypothesis test at the significance levelα, the null hypothesisH0:μ1=μ2 will be rejected in favor of the al- ternative hypothesisHa:μ1< μ2if and only if the (1−α)-level upper confidence bound forμ1−μ2 is negative. In each case,
464 CHAPTER 10 Inferences for Two Population Means illustrate the preceding relationship by obtaining the appropriate upper confidence bound and comparing the result to the conclu- sion of the hypothesis test in the specified exercise.
a. Exercise 10.71 b. Exercise 10.72
10.93 Right-Tailed Hypothesis Tests and CIs. If the as- sumptions for a nonpooledt-interval are satisfied, the formula for a (1−α)-level lower confidence bound for the difference, μ1−μ2, between two population means is
(x¯1− ¯x2)−tαã
(s12/n1)+(s22/n2).
For a right-tailed hypothesis test at the significance levelα, the null hypothesis H0:μ1=μ2will be rejected in favor of the al- ternative hypothesisHa:μ1> μ2if and only if the (1−α)-level lower confidence bound forμ1−μ2is positive. In each case, il- lustrate the preceding relationship by obtaining the appropriate lower confidence bound and comparing the result to the conclu- sion of the hypothesis test in the specified exercise.
a. Exercise 10.70 b. Exercise 10.73