JERZY NEYMAN: A PRINCIPAL FOUNDER OF MODERN STATISTICAL THEORY
put 10.4. Steps for generating that output are presented in Instructions 10.4. Note
As shown in Output 10.4, the P-value for the hypothesis test is about 0.048.
Because the P-value is less than the specified significance level of 0.05, we re- jectH0. Output 10.4 also shows that a 95% confidence interval for the difference between the means is from 0.04 to 7.16.
MINITAB
OUTPUT 10.4 Pairedt-procedures on the age data
10.5 Inferences for Two Population Means, Using Paired Samples 515
EXCEL
OUTPUT 10.4 (cont.) Pairedt-procedures on the age data
TI-83/84 PLUS
UsingT-Test
UsingTInterval
INSTRUCTIONS 10.4 Steps for generating Output 10.4 MINITAB
1 Store the age data from the second and third columns of Table 10.13 in columns named HUSBAND and WIFE 2 ChooseStat➤Basic Statistics➤Paired t. . .
3 Press the F3 key to reset the dialog box
4 Click in theSample 1text box and specify HUSBAND 5 Click in theSample 2text box and specify WIFE 6 Click theOptions. . . button
7 Click in theConfidence leveltext box and type95 8 Click the arrow button at the right of theAlternative
hypothesisdrop-down list box and selectDifference= hypothesized difference
9 ClickOKtwice EXCEL
1 Store the age data from the second and third columns of Table 10.13 in columns named HUSBAND and WIFE 2 ChooseXLSTAT➤Parametric tests➤Two-sample
t-test and z-test
3 Click the reset button in the lower left corner of the dialog box
4 Click in theSample 1selection box and then select the column of the worksheet that contains the HUSBAND data
5 Click in theSample 2selection box and then select the column of the worksheet that contains the WIFE data 6 In theData formatlist, select thePaired samples
option button 7 Click theOptionstab
8 Click the arrow button at the right of theAlternative hypothesisdrop-down list box and selectMean 1 – Mean 2=D
9 Type5in theSignificance level (%)text box
10 ClickOK
11 Click theContinuebutton in theXLSTAT – Selections dialog box
TI-83/84 PLUS
Store the age data from the second and third columns of Table 10.13 in lists named HUSB and WIFE.
FOR THE PAIRED DIFFERENCES:
1 Press2nd➤LIST, arrow down to HUSB, and press ENTER
2 Press−
3 Press2nd➤LIST, arrow down to WIFE, and pressENTER 4 PressSTO➧
5 Press2nd➤A-LOCK, typeDIFF, and pressENTER FOR THE HYPOTHESIS TEST:
1 PressSTAT, arrow over toTESTS, and press2 2 HighlightDataand pressENTER
3 Press the down-arrow key, type0forμ0, and press ENTER
4 Press2nd➤LIST, arrow down to DIFF, and press ENTERtwice
5 Type1forFreqand then pressENTER 6 Highlight=μ0and pressENTER
7 Arrow down toCalculateand pressENTER FOR THE CI:
1 PressSTAT, arrow over toTESTS, and press8 2 HighlightDataand pressENTER
3 Press the down-arrow key
4 Press2nd➤LIST, arrow down to DIFF, and press ENTERtwice
5 Type1forFreqand then pressENTER 6 Type.95forC-Leveland pressENTERtwice
Note to Minitab and Excel users: As we have noted, Minitab and Excel compute a two-sided confidence interval for a two-tailed test and a one-sided confidence interval for a one-tailed test. To perform a one-tailed hypothesis test and obtain a two-sided confidence interval, apply Minitab’s or Excel’s pairedt-procedure twice: once for the one-tailed hypothesis test and once for the confidence interval specifying a two-tailed hypothesis test.
Note to TI-83/84 Plus users:The pairedt-procedures are just one-meant-procedures applied to the paired differences. Since, at the time of this writing, the TI-83/84 Plus does not have built-in pairedt-procedures, we applied its one-meant-procedures to the paired differences, as seen in Instructions 10.4.
Exercises 10.5
Understanding the Concepts and Skills
10.139 State one possible advantage of using paired samples instead of independent samples.
10.140 What constitutes each pair in a paired sample?
10.141 State the two conditions required for performing a paired t-procedure. How important are those conditions?
10.142 Provide an example (different from the ones considered in this section) of a procedure based on a paired sample being more ap- propriate than one based on independent samples.
In Exercises10.143–10.148, hypothesis tests are proposed. For each hypothesis test,
a. identify the variable.
b. identify the two populations.
c. identify the pairs.
d. identify the paired-difference variable.
e. determine the null and alternative hypotheses.
f. classify the hypothesis test as two tailed, left tailed, or right tailed.
10.143 TV Viewing. TheA. C. Nielsen Companycollects data on the TV viewing habits of Americans and publishes the information in Nielsen Report on Television. Suppose that you want to use a paired sample to decide whether the mean viewing times of married men and married women differ.
10.144 Self-Reported Weight. The article “Accuracy of Self- Reported Height and Weight in a Community-Based Sample of Older African Americans and Whites” (Journal of Gerontology Se- ries A: Biological Sciences and Medical Sciences, Vol. 65A, No. 10, pp. 1123–1129) by G. Fillenbaum et al. explores the relation- ship between measured and self-reported height and weight. The authors sampled African American and White women and men older than 70 years of age. A hypothesis test is to be performed to decide whether, on average, self-reported weight is less than measured weight for the aforementioned age group.
10.145 Hypnosis and Pain. In the paper “An Analysis of Factors That Contribute to the Efficacy of Hypnotic Analgesia” (Journal of Abnormal Psychology, Vol. 96, No. 1, pp. 46–51), D. Price and J. Barber examined the effects of hypnosis on pain. They measured response to pain using a visual analogue scale (VAS), in centimeters, where higher VAS indicates greater pain. VAS sensory ratings were made before and after hypnosis on each of 16 subjects. A hypothe- sis test is to be performed to decide whether, on average, hypnosis reduces pain.
10.146 Sports Stadiums and Home Values. In the paper “Hous- ing Values Near New Sporting Stadiums” (Land Economics, Vol. 81, Issue 3, pp. 379–395), C. Tu examined the effects of construction of new sports stadiums on home values. Suppose that you want to use a paired sample to decide whether construction of new sports stadiums affects the mean price of neighboring homes.
10.147 Breastmilk and Antioxidants. There is convincing evi- dence that breastmilk containing antioxidants is important in the pre- vention of diseases in infants. Researchers A. Xavier et al. studied the effects of storing breastmilk on antioxidant levels in the article “Total Antioxidant Concentrations of Breastmilk—An Eye-Opener to the Negligent” (Journal of Health, Population and Nutrition, Vol. 29, No. 6, pp. 605–611). Samples of breastmilk were taken from women and divided into fresh samples that were immediately tested and the remaining samples that were stored in the refrigerator and tested after 48 hours. A hypothesis test is to be performed to decide whether, on average, stored breastmilk has a lower total antioxidant capacity.
10.148 Fiber Density. In the article “Comparison of Fiber Counting by TV Screen and Eyepieces of Phase Contrast Microscopy” (Amer- ican Industrial Hygiene Association Journal, Vol. 63, pp. 756–761), I. Moa et al. reported on determining fiber density by two different methods. The fiber density of 10 samples with varying fiber density was obtained by using both an eyepiece method and a TV-screen method. A hypothesis test is to be performed to decide whether, on average, the eyepiece method gives a greater fiber density reading than the TV-screen method.
In Exercises10.149–10.154, the null hypothesis is H0:μ1=μ2and the alternative hypothesis is as specified. We have provided data from a simple random paired sample from the two populations under con- sideration. In each case, use the paired t-test to perform the required hypothesis test at the 10% significance level.
10.149 Ha:μ1=μ2
Observation from Pair Population 1 Population 2
1 13 11
2 16 15
3 13 10
4 14 8
5 12 8
6 8 9
7 17 14
10.5 Inferences for Two Population Means, Using Paired Samples 517 10.150 Ha:μ1< μ2
Observation from Pair Population 1 Population 2
1 7 13
2 4 9
3 10 6
4 0 2
5 20 19
6 −1 5
7 12 10
10.151 Ha:μ1> μ2
Observation from Pair Population 1 Population 2
1 7 6
2 9 8
3 12 11
4 15 14
5 27 6
6 16 9
7 11 5
8 8 2
10.152 Ha:μ1=μ2
Observation from Pair Population 1 Population 2
1 10 12
2 8 7
3 13 11
4 13 16
5 17 15
6 12 9
7 12 12
8 11 7
10.153 Ha:μ1< μ2
Observation from Pair Population 1 Population 2
1 15 18
2 22 25
3 15 17
4 27 24
5 24 30
6 23 23
7 8 10
8 20 27
9 2 3
10.154 Ha:μ1> μ2
Observation from Pair Population 1 Population 2
1 40 32
2 30 29
3 34 36
4 22 18
5 35 31
6 26 26
7 26 25
8 27 25
9 11 15
10 35 31
Applying the Concepts and Skills
Preliminary data analyses indicate that use of a paired t-test is rea- sonable in Exercises10.155–10.160. Perform each hypothesis test by using either the critical-value approach or the P-value approach.
10.155 Behavioral Genetics. In the article “Growth references for height, weight and BMI of Twins aged 0–2.5 years” (ACTA Pedi- atrica, Vol. 97, pp. 1099–1104), the researchers P. Dommelen et al.
determined the size of the growth deficit in Dutch monozygotic and dizygotic twins aged between 0–2.5 years as compared to the single- tons and to construct reference growth charts for twins. The follow- ing table shows the difference of the height of the twins at various age groups from 0 to 2.5 years.
5.2 4.5 3.2 4.1 4.8 5.0 4.3 2.7 3.4 3.8 4.7 3.9 1.9 2.3 1.0 a. Identify the variable under consideration.
b. Identify the two populations.
c. Identify the paired-difference variable.
d. Are the numbers in the table paired differences? Why or why not?
e. At the 5% significance level, do the data provide sufficient evi- dence to conclude that the mean heights of twins separately differ?
(Note: d=3.65 cm andsd =1.23 cm.) f. Repeat part (e) at the 1% significance level.
10.156 Sleep. In 1908, W. S. Gosset published “The Probable Error of a Mean” (Biometrika, Vol. 6, pp. 1–25). In this pioneering paper, published under the pseudonym “Student,” he introduced what later became known as Student’st-distribution. Gosset used the following data set, which gives the additional sleep in hours obtained by 10 patients who used laevohysocyamine hydrobromide.
1.9 0.8 1.1 0.1 −0.1
4.4 5.5 1.6 4.6 3.4
a. Identify the variable under consideration.
b. Identify the two populations.
c. Identify the paired-difference variable.
d. Are the numbers in the table paired differences? Why or why not?
e. At the 5% significance level, do the data provide sufficient evi- dence to conclude that laevohysocyamine hydrobromide is effec- tive in increasing sleep? (Note: d=2.33 andsd=2.002.) f. Repeat part (e) at the 1% significance level.
10.157 Anorexia Treatment. Anorexia nervosa is a serious eating disorder, particularly among young women. The following data pro- vide the weights, in pounds, of 17 anorexic young women before and after receiving a family therapy treatment for anorexia nervosa.
[SOURCE: D. Hand et al. (ed.)A Handbook of Small Data Sets, Lon- don: Chapman & Hall, 1994; raw data from B. Everitt (personal com- munication)]
Before After Before After Before After
83.3 94.3 76.9 76.8 82.1 95.5
86.0 91.5 94.2 101.6 77.6 90.7
82.5 91.9 73.4 94.9 83.5 92.5
86.7 100.3 80.5 75.2 89.9 93.8
79.6 76.7 81.6 77.8 86.0 91.7
87.3 98.0 83.8 95.2
Does family therapy appear to be effective in helping anorexic young women gain weight? Perform the appropriate hypothesis test at the 5% significance level.
10.158 Measuring Treadwear. R. Stichler et al. compared two methods of measuring treadwear in their paper “Measurement of Treadwear of Commercial Tires” (Rubber Age, Vol. 73:2). Eleven tires were each measured for treadwear by two methods, one based on weight and the other on groove wear. The data, in thousands of miles, are as follows.
Weight Groove Weight Groove method method method method
30.5 28.7 24.5 16.1
30.9 25.9 20.9 19.9
31.9 23.3 18.9 15.2
30.4 23.1 13.7 11.5
27.3 23.7 11.4 11.2
20.4 20.9
At the 5% significance level, do the data provide sufficient evidence to conclude that, on average, the two measurement methods give dif- ferent results?
10.159 Glaucoma and Corneal Thickness. Glaucoma is a lead- ing cause of blindness in the United States. N. Ehlers measured the corneal thickness of eight patients who had glaucoma in one eye but not in the other. The results of the study were published as the paper
“On Corneal Thickness and Intraocular Pressure, II” (Acta Opthal- mologica, Vol. 48, pp. 1107–1112). The data on corneal thickness, in microns, are shown in the following table.
Patient Normal Glaucoma
1 484 488
2 478 478
3 492 480
4 444 426
5 436 440
6 398 410
7 464 458
8 476 460
At the 10% significance level, do the data provide sufficient evidence to conclude that mean corneal thickness is greater in normal eyes than in eyes with glaucoma?
10.160 Cooling Down. Cooling down with a cold drink before ex- ercise in the heat is believed to help an athlete perform. Researcher J. Dugas explored the difference between cooling down with an ice slurry (slushy) and with cold water in the article “Ice Slurry Inges- tion Increases Running Time in the Heat” (Clinical Journal of Sports Medicine, Vol. 21, No. 6, pp. 541–542). Ten male participants drank a flavored ice slurry and ran on a treadmill in a controlled hot and hu- mid environment. Days later, the same participants drank cold water and ran on a treadmill in the same hot and humid environment. The following table shows the times, in minutes, it took to fatigue on the treadmill for both the ice slurry and the cold water.
Subject Cold Water Ice Slurry
1 52 56
2 37 43
3 44 52
4 51 58
5 34 38
6 38 45
7 41 45
8 50 58
9 29 34
10 38 44
At the 1% significance level, do the data provide sufficient evidence to conclude that, on average, cold water is less effective than ice slurry for optimizing athletic performance in the heat? (Note:The mean and standard deviation of the paired differences are−5.9 minutes and 1.60 minutes, respectively.)
In Exercises10.161–10.166, apply Procedure 10.7 on page 513 to obtain the required confidence interval. Interpret your result in each case.
10.161 Behavioral Genetics. Refer to Exercise 10.155.
a. Determine a 95% confidence interval for the difference between the mean heights of the twins.
b. Repeat part (a) for a 99% confidence level.
10.162 Sleep. Refer to Exercise 10.156.
a. Determine a 90% confidence interval for the additional sleep that would be obtained, on average, by using laevohysocyamine hy- drobromide.
b. Repeat part (a) for a 98% confidence level.
10.163 Anorexia Treatment. Refer to Exercise 10.157 and find a 90% confidence interval for the weight gain that would be obtained, on average, by using the family therapy treatment.
10.164 Measuring Treadwear. Refer to Exercise 10.158 and find a 95% confidence interval for the mean difference in measurement by the weight and groove methods.
10.165 Glaucoma and Corneal Thickness. Refer to Exer- cise 10.159 and obtain an 80% confidence interval for the difference between the mean corneal thickness of normal eyes and that of eyes with glaucoma.
10.166 Cooling Down. Refer to Exercise 10.160 and find a 98% confidence interval for the difference between the mean times to fatigue on a treadmill in a hot and humid environment after cool- ing down with cold water and after cooling down with an ice slurry.
10.5 Inferences for Two Population Means, Using Paired Samples 519
In each of Exercises10.167–10.169, use the technology of your choice to perform the required tasks.
10.167 Font Readability. In the online paper “A Comparison of Two Computer Fonts: Serif versus Ornate Sans Serif” (Usability News, Issue 5.3), researchers S. Morrison and J. Noyes studied whether the type of font used in a document affects reading speed or comprehension. The fonts used for the comparisons were the serif font Times New Roman (TNR) and a more ornate sans serif font called Gigi. The following table gives the times, in seconds, that it took each of the 25 participants to read paragraphs in the TNR and Gigi fonts.
TNR Gigi TNR Gigi TNR Gigi
27.00 32.88 16.43 18.15 34.20 37.83 23.20 23.00 20.75 27.18 25.01 29.90 23.00 37.87 31.41 34.00 12.93 21.82 33.76 43.53 24.69 26.50 15.84 20.41 19.42 21.06 24.60 27.00 21.03 27.03 15.56 23.00 22.42 29.30 26.23 32.03 19.13 23.40 18.41 22.24 13.60 20.02 19.41 24.49 24.56 31.44
22.28 22.76 16.75 17.95
Suppose that you want to perform a hypothesis test to determine whether, on average, people read faster with the TNR font than with the Gigi font. Conduct preliminary graphical data analyses to de- cide whether applying the pairedt-test is reasonable. Explain your decision.
10.168 Tobacco Mosaic Virus. To assess the effects of two differ- ent strains of the tobacco mosaic virus, W. Youden and H. Beale ran- domly selected eight tobacco leaves. Half of each leaf was subjected to one of the strains of tobacco mosaic virus and the other half to the other strain. The researchers then counted the number of local lesions apparent on each half of each leaf. The results of their study were pub- lished in the paper “A Statistical Study of the Local Lesion Method for Estimating Tobacco Mosaic Virus” (Contributions to Boyce Thomp- son Institute, Vol. 6, p. 437). Here are the data.
Leaf 1 2 3 4 5 6 7 8
Virus 1 31 20 18 17 9 8 10 7
Virus 2 18 17 14 11 10 7 5 6
Suppose that you want to perform a hypothesis test to determine whether a difference exists between the mean numbers of local lesions resulting from the two viral strains. Conduct preliminary graphical analyses to decide whether applying the pairedt-test is rea- sonable. Explain your decision.
10.169 Antiviral Therapy. In the article “Improved Outcome for Children With Disseminated Adenoviral Infection Following Allo- geneic Stem Cell Transplantation” (British Journal of Haematol- ogy, Vol. 130, Issue 4, p. 595), B. Kampmann et al. examined chil- dren who received stem cell transplants and subsequently became in- fected with a variety of ailments. A new antiviral therapy was ad- ministered to 11 patients. Their absolute lymphocyte counts (ABS lymphs) (×109/L) at onset and resolution were as shown in the fol- lowing table.
Onset Resolution Onset Resolution
0.08 0.59 0.31 0.38
0.02 0.37 0.23 0.39
0.03 0.07 0.09 0.02
0.64 0.81 0.10 0.38
0.03 0.76 0.04 0.60
0.15 0.44
a. Obtain normal probability plots and boxplots of the onset data, the resolution data, and the paired differences of those data.
b. Based on your results from part (a), is applying a one-mean t-procedure to the onset data reasonable?
c. Based on your results from part (a), is applying a one-mean t-procedure to the resolution data reasonable?
d. Based on your results from part (a), is applying a paired t-procedure to the data reasonable?
e. What do your answers from parts (b)–(d) imply about the condi- tions for using a pairedt-procedure?
Working with Large Data Sets
10.170 Faculty Salaries. TheAmerican Association of University Professors(AAUP) conducts salary studies of college professors and publishes its findings inAAUP Annual Report on the Economic Sta- tus of the Profession. In Example 10.3 on pages 471–473, we per- formed a hypothesis test based on independent samples to decide whether mean salaries differ for faculty in private and public insti- tutions. Now you are to perform that same hypothesis test based on a paired sample. Pairs were formed by matching faculty in private and public institutions by rank and specialty. A random sample of 30 pairs yielded the data, in thousands of dollars, presented on the WeissStats site. Use the technology of your choice to do the following tasks.
a. Decide, at the 5% significance level, whether the data provide sufficient evidence to conclude that mean salaries differ for fac- ulty in private institutions and public institutions. Use the paired t-test.
b. Compare your result in part (a) to the one obtained in Exam- ple 10.3.
c. Repeat both the pooledt-test of Example 10.3 and the paired t-test of part (a), using a 1% significance level, and compare your results.
d. Which test do you think is preferable here: the pooledt-test or the pairedt-test? Explain your answer.
e. Find and interpret a 95% confidence interval for the difference between the mean salaries of faculty in private and public institu- tions. Use the pairedt-interval procedure.
f. Compare your result in part (e) to the one obtained in Exam- ple 10.4 on page 474.
g. Obtain a normal probability plot and a boxplot of the paired dif- ferences.
h. Based on your graphs from part (g), do you think that applying pairedt-procedures here is reasonable?
10.171 Marriage Ages. In the Statistics Norway on-line article
“The Times They Are a Changing,” J. Kristiansen discussed the changes in age at the time of marriage in Norway. The ages, in years, at the time of marriage for 75 Norwegian couples are presented on the WeissStats site. Use the technology of your choice to do the following.
a. Decide, at the 1% significance level, whether the data provide suf- ficient evidence to conclude that the mean age of Norwegian men at the time of marriage exceeds that of Norwegian women.
b. Find and interpret a 99% confidence interval for the difference be- tween the mean ages at the time of marriage for Norwegian men and women.
c. Remove the two paired-difference (potential) outliers and repeat parts (a) and (b). Compare your results to those in parts (a) and (b).
10.172 Storm Hydrology and Clear Cutting. In the document
“Peak Discharge from Unlogged and Logged Watersheds,” J. Jones and G. Grant compiled (paired) data on peak discharge from storms in two watersheds, one unlogged and one logged (100% clear-cut). If there is an effect due to clear-cutting, one would expect that the runoff would be greater in the logged area than in the unlogged area. The runoffs, in cubic meters per second per square kilometer (m3/s/ km2), are provided on the WeissStats site. Use the technology of your choice to do the following.
a. Formulate the null and alternative hypotheses to reflect the expec- tation expressed above.
b. Perform the required hypothesis test at the 1% significance level.
c. Obtain and interpret a 99% confidence interval for the difference between mean runoffs in the logged and unlogged watersheds.
d. Construct a histogram of the sample data to identify the approxi- mate shape of the paired-difference variable.
e. Based on your result from part (d), do you think that applying the pairedt-procedures in parts (b) and (c) is reasonable? Explain your answer.
Extending the Concepts and Skills
10.173 Explain exactly how a pairedt-test can be formulated as a one- meant-test. (Hint:Work solely with the paired-difference variable.) 10.174 A hypothesis test, based on a paired sample, is to be per- formed to compare the means of two populations. The sample of 15 paired differences contains an outlier but otherwise is approxi- mately bell shaped. Assuming that removal of the outlier would not be legitimate, would use of the pairedt-test or a nonparametric test be better? Explain your answer.
10.175 Gasoline Additive. This exercise shows what can hap- pen when a hypothesis-testing procedure designed for use with independent samples is applied to perform a hypothesis test on a paired sample. The gas mileages, in miles per gallon (mpg), of 10 ran- domly selected cars, both with and without a new gasoline additive, are shown in the following table.
With additive Without additive
25.7 24.9
20.0 18.8
28.4 27.7
13.7 13.0
18.8 17.8
12.5 11.3
28.4 27.8
8.1 8.2
23.1 23.1
10.4 9.9
a. Apply the paired t-test to decide, at the 5% significance level, whether the gasoline additive is effective in increasing gas mileage.
b. Apply the pooledt-test to the sample data to perform the hypoth- esis test.
c. Why is performing the hypothesis test the way you did in part (b) inappropriate?
d. Compare your result in parts (a) and (b).
10.176 Permutation Tests. With the advent of high-speed comput- ing, new procedures have been developed that permit statistical in- ferences to be performed under less restrictive conditions than those of classical procedures.Permutation testsconstitute one such col- lection of new procedures. To perform a permutation test to com- pare two population means using paired samples, we proceed as follows.
1. For each pair, switch or don’t switch the two observations with probability 0.5. This procedure yields a new paired sample.
2. Compute the mean of the new paired differences.
3. Repeat steps 1 and 2 a large number (hundreds or thousands) of times.
4. The distribution of the resulting paired-difference means provides an estimate of the sampling distribution of the sample mean of paired differences when the null hypothesis of equal population means is true. This estimate is called apermutation distribution.
5. The (estimated)P-value of the hypothesis test equals the propor- tion of values of the permutation distribution that are as extreme as or more extreme than the observed mean of the paired differences.
Refer to Example 10.16 on pages 511–512. Use the technology of your choice to conduct a permutation test and compare your results with those found by using the pairedt-test. Discuss any discrepancy that you encounter.
10.6 The Paired Wilcoxon Signed-Rank Test∗
In Section 10.5, we discussed the pairedt-procedures, which provide methods for com- paring two population means using paired samples. An assumption for use of those pro- cedures is that the paired-difference variable is (approximately) normally distributed or that the sample size is large. For a small or moderate sample size where the dis- tribution of the paired-difference variable is far from normal, a pairedt-procedure is inappropriate and a nonparametric procedure should be used instead.
For instance, if the distribution of the paired-difference variable is symmetric (but not necessarily normal), we can perform a hypothesis test to compare the means of the two populations by applying the Wilcoxon signed-rank test (Procedure 9.3 on page 434)