Two-Sample Case Study

10.14 Two-Sample Case Study

In this section, we consider a study involving a thorough graphical and formal anal- ysis, along with annotated computer printout and conclusions. In a data analysis study conducted by personnel at the Statistics Consulting Center at Virginia Tech, two different materials, alloy A and alloy B, were compared in terms of breaking strength. Alloy B is more expensive, but it should certainly be adopted if it can

be shown to be stronger than alloy A. The consistency of performance of the two alloys should also be taken into account. Random samples of beams made from each alloy were selected, and strength was measured in units of 0.001-inch deflection as a fixed force was applied at both ends of the beam. Twenty specimens were used for each of the two alloys. The data are given in Table 10.13.

It is important that the engineer compare the two alloys. Of concern is average strength and reproducibility. It is of interest to determine if there is a severe

380 Chapter 10 One- and Two-Sample Tests of Hypotheses

Table 10.13: Data for Two-Sample Case Study

Alloy A

Alloy B

violation of the normality assumption required of both the t- and F -tests. Figures

10.21 and 10.22 are normal quantile-quantile plots of the samples of the two alloys. There does not appear to be any serious violation of the normality assumption. In addition, Figure 10.23 shows two box-and-whisker plots on the same graph. The box-and-whisker plots suggest that there is no appreciable difference in the vari- ability of deflection for the two alloys. However, it seems that the mean deflection for alloy B is significantly smaller, suggesting, at least graphically, that alloy B is stronger. The sample means and standard deviations are

y ¯ B = 79.70, s B = 3.097. The SAS printout for the PROC TTEST is shown in Figure 10.24. The F -test

¯ y A = 83.55, s A = 3.663;

suggests no significant difference in variances (P = 0.4709), and the two-sample t-statistic for testing

H 0 :μ A =μ B ,

H 1 :μ A >μ B

(t = 3.59, P = 0.0009) rejects H 0 in favor of H 1 and thus confirms what the graphical information suggests. Here we use the t-test that pools the two-sample variances together in light of the results of the F -test. On the basis of this analysis, the adoption of alloy B would seem to be in order.

Statistical Significance and Engineering or Scientific Significance

While the statistician may feel quite comfortable with the results of the comparison between the two alloys in the case study above, a dilemma remains for the engineer. The analysis demonstrated a statistically significant improvement with the use of alloy B. However, is the difference found really worth pursuing, since alloy

B is more expensive? This illustration highlights a very important issue often overlooked by statisticians and data analysts—the distinction between statistical significance and engineering or scientific significance. Here the average difference

in deflection is ¯ y A − ¯y B = 0.00385 inch. In a complete analysis, the engineer must determine if the difference is sufficient to justify the extra cost in the long run. This is an economic and engineering issue. The reader should understand that a statistically significant difference merely implies that the difference in the sample

10.14 Two-Sample Case Study 381

Normal Quantile Normal Quantile Figure 10.21: Normal quantile-quantile plot of

Figure 10.22: Normal quantile-quantile plot of data for alloy A.

data for alloy B.

Alloy A

Alloy B

Figure 10.23: Box-and-whisker plots for both alloys.

means found in the data could hardly have occurred by chance. It does not imply that the difference in the population means is profound or particularly significant in the context of the problem. For example, in Section 10.4, an annotated computer printout was used to show evidence that a pH meter was, in fact, biased. That is, it does not demonstrate a mean pH of 7.00 for the material on which it was tested. But the variability among the observations in the sample is very small. The engineer may decide that the small deviations from 7.0 render the pH meter adequate.

382 Chapter 10 One- and Two-Sample Tests of Hypotheses

The TTEST Procedure

Alloy

Mean Std Dev Std Err

Alloy A

Alloy B

DF t Value

Equality of Variances

Num DF

Den DF

F Value

Pr > F

19 19 1.40 0.4709 Figure 10.24: Annotated SAS printout for alloy data.

Exercises

10.79 A machine is supposed to mix peanuts, hazel- periment 256 times, we obtained the following results: nuts, cashews, and pecans in the ratio 5:2:2:1. A can containing 500 of these mixed nuts was found to have

1 2 3 4 5 6 7 8 269 peanuts, 112 hazelnuts, 74 cashews, and 45 pecans.

60 34 12 9 1 3 1 At the 0.05 level of significance, test the hypothesis Test the hypothesis, at the 0.05 level of significance,

f 136

that the machine is mixing the nuts in the ratio 5:2:2:1. that the observed distribution of X may be fitted by the geometric distribution g(x; 1/2), x = 1, 2, 3, . . . .

10.80 The grades in a statistics course for a particu- lar semester were as follows:

10.84 For Exercise 1.18 on page 31, test the good- Grade

A B C D F ness of fit between the observed class frequencies and f 14 18 32 20 16 the corresponding expected frequencies of a normal dis- tribution with μ = 65 and σ = 21, using a 0.05 level of

Test the hypothesis, at the 0.05 level of significance, significance. that the distribution of grades is uniform.

10.85 For Exercise 1.19 on page 31, test the good- 10.81 A die is tossed 180 times with the following

results: ness of fit between the observed class frequencies and the corresponding expected frequencies of a normal dis-

x 1 2 3 4 5 6 tribution with μ = 1.8 and σ = 0.4, using a 0.01 level f 28 36 36 30 27 23 of significance.

Is this a balanced die? Use a 0.01 level of significance. 10.86 In an experiment to study the dependence of hypertension on smoking habits, the following data

10.82 Three marbles are selected from an urn con- were taken on 180 individuals: taining 5 red marbles and 3 green marbles. After the number X of red marbles is recorded, the marbles are

Non- Moderate Heavy replaced in the urn and the experiment repeated 112

smokers Smokers Smokers times. The results obtained are as follows:

21 36 30 x

Hypertension

48 26 19 f 1 31 55 25 Test the hypothesis that the presence or absence of hy-

0 1 2 3 No hypertension

pertension is independent of smoking habits. Use a Test the hypothesis, at the 0.05 level of significance,

0.05 level of significance.

that the recorded data may be fitted by the hypergeo- metric distribution h(x; 8, 3, 5), x = 0, 1, 2, 3.

10.87 A random sample of 90 adults is classified ac- cording to gender and the number of hours of television

10.83 A coin is thrown until a head occurs and the watched during a week: number X of tosses recorded. After repeating the ex- 10.83 A coin is thrown until a head occurs and the watched during a week: number X of tosses recorded. After repeating the ex-

in Across the Board (June 1981): Over 25 hours

Under 25 hours 27 19 Standard of Living Somewhat

Not as Use a 0.01 level of significance and test the hypothesis

Same Good Total that the time spent watching television is independent of whether the viewer is male or female.

63 135 102 300 10.88 A random sample of 200 married men, all re-

May

47 100 53 200 tired, was classified according to education and number 1981: Jan.

Sept.

40 105 55 200 of children:

Test the hypothesis that the proportions of households Number of Children

within each standard of living category are the same Education

for each of the four time periods. Use a P -value. Elementary

Secondary 19 42 17 10.92 A college infirmary conducted an experiment College

12 17 10 to determine the degree of relief provided by three cough remedies. Each cough remedy was tried on 50

Test the hypothesis, at the 0.05 level of significance, students and the following data recorded: that the size of a family is independent of the level of education attained by the father.

Cough Remedy NyQuil Robitussin Triaminic

11 13 9 mine whether the incidence of certain types of crime

10.89 A criminologist conducted a survey to deter-

No relief

32 28 27 varied from one part of a large city to another. The

Some relief

7 9 14 particular crimes of interest were assault, burglary, Test the hypothesis that the three cough remedies are larceny, and homicide. The following table shows the equally effective. Use a P -value in your conclusion. numbers of crimes committed in four areas of the city during the past year.

Total relief

10.93 To determine current attitudes about prayer Type of Crime

in public schools, a survey was conducted in four Vir- District Assault Burglary Larceny Homicide

ginia counties. The following table gives the attitudes 1 162

18 of 200 parents from Craig County, 150 parents from 2 310

25 Giles County, 100 parents from Franklin County, and 3 258

10 100 parents from Montgomery County: 4 280

19 County Can we conclude from these data at the 0.01 level of

Attitude Craig Giles Franklin Mont. significance that the occurrence of these types of crime

65 66 40 34 is dependent on the city district?

93 54 27 24 10.90 According to a Johns Hopkins University study Test for homogeneity of attitudes among the four coun-

No opinion

published in the American Journal of Public Health, ties concerning prayer in the public schools. Use a P - widows live longer than widowers. Consider the fol- value in your conclusion. lowing survival data collected on 100 widows and 100 widowers following the death of a spouse:

10.94 A survey was conducted in Indiana, Kentucky, and Ohio to determine the attitude of voters concern-

Years Lived Widow

ing school busing. A poll of 200 voters from each of Less than 5

Widower

25 39 these states yielded the following results: 5 to 10

More than 10 33 21 Voter Attitude Do Not

Support Support Undecided the proportions of widows and widowers are equal with respect to the different time periods that a spouse sur-

Can we conclude at the 0.05 level of significance that

State

82 97 21 vives after the death of his or her mate?

93 74 33 10.91 The following responses concerning the stan- At the 0.05 level of significance, test the null hypothe-

Ohio

dard of living at the time of an independent opinion sis that the proportions of voters within each attitude poll of 1000 households versus one year earlier seem to category are the same for each of the three states.

384 Chapter 10 One- and Two-Sample Tests of Hypotheses 10.95 A survey was conducted in two Virginia cities esis that proportions of voters favoring candidate A,

to determine voter sentiment about two gubernatorial favoring candidate B, and undecided are the same for candidates in an upcoming election. Five hundred vot- each city. ers were randomly selected from each city and the fol- lowing data were recorded:

10.96 In a study to estimate the proportion of wives who regularly watch soap operas, it is found that 52

of 200 wives in Denver, 31 of 150 wives in Phoenix, Voter Sentiment

City

and 37 of 150 wives in Rochester watch at least one Favor A

Richmond

Norfolk

soap opera. Use a 0.05 level of significance to test the Favor B

hypothesis that there is no difference among the true Undecided

85 77 proportions of wives who watch soap operas in these At the 0.05 level of significance, test the null hypoth- three cities.

Review Exercises

With CO Without CO used in testing the following claims and determine gen-

10.97 State the null and alternative hypotheses to be

Subject

1 26.46 25.41 erally where the critical region is located:

2 17.46 22.53 (a) The mean snowfall at Lake George during the

3 16.32 16.32 month of February is 21.8 centimeters.

(b) No more than 20% of the faculty at the local uni- 6 20.65 21.77 versity contributed to the annual giving fund.

7 28.21 28.17 (c) On the average, children attend schools within 6.2

8 33.94 32.02 kilometers of their homes in suburban St. Louis.

9 29.32 28.96 (d) At least 70% of next year’s new cars will be in the It is conjectured that oxygen consumption should be compact and subcompact category.

higher in an environment relatively free of CO. Do a (e) The proportion of voters favoring the incumbent in significance test and discuss the conjecture. the upcoming election is 0.58.

10.101 In a study analyzed by the Statistics Consult- (f) The average rib-eye steak at the Longhorn Steak ing Center at Virginia Tech, a group of subjects was house weighs at least 340 grams.

asked to complete a certain task on the computer. The response measured was the time to completion. The

10.98 A geneticist is interested in the proportions of purpose of the experiment was to test a set of facilita- males and females in a population who have a cer- tion tools developed by the Department of Computer tain minor blood disorder. In a random sample of 100 Science at the university. There were 10 subjects in- males, 31 are found to be afflicted, whereas only 24 of volved. With a random assignment, five were given a 100 females tested have the disorder. Can we conclude standard procedure using Fortran language for comple- at the 0.01 level of significance that the proportion of tion of the task. The other five were asked to do the men in the population afflicted with this blood disorder task with the use of the facilitation tools. The data on is significantly greater than the proportion of women the completion times for the task are given here. afflicted?

Group 2 10.99 A study was made to determine whether more

Group 1

(Standard Procedure) (Facilitation Tool) Italians than Americans prefer white champagne to

132 pink champagne at weddings. Of the 300 Italians

162 selected at random, 72 preferred white champagne,

134 and of the 400 Americans selected, 70 preferred white

138 champagne. Can we conclude that a higher proportion

133 of Italians than Americans prefer white champagne at Assuming that the population distributions are nor- weddings? Use a 0.05 level of significance.

mal and variances are the same for the two groups, 10.100 Consider the situation of Exercise 10.54 on support or refute the conjecture that the facilitation

page 360. Oxygen consumption in mL/kg/min, was tools increase the speed with which the task can be also measured.

accomplished. 10.102 State the null and alternative hypotheses to

be used in testing the following claims, and determine

Review Exercises 385 generally where the critical region is located:

nificant change in WBC leukocytes with the surgery. (a) At most, 20% of next year’s wheat crop will be

exported to the Soviet Union. 10.106 A study was conducted at the Department of (b) On the average, American homemakers drink 3 Health and Physical Education at Virginia Tech to de- cups of coffee per day.

termine if 8 weeks of training truly reduces the choles- terol levels of the participants. A treatment group con-

(c) The proportion of college graduates in Virginia this sisting of 15 people was given lectures twice a week year who majored in the social sciences is at least on how to reduce cholesterol level. Another group of 0.15.

18 people of similar age was randomly selected as a (d) The average donation to the American Lung Asso- control group. All participants’ cholesterol levels were ciation is no more than $10.

recorded at the end of the 8-week program and are (e) Residents in suburban Richmond commute, on the listed below. average, 15 kilometers to their place of employ- Treatment:

ment. 129 131 154 172 115 126 175 191 10.103 If one can containing 500 nuts is selected

122 238 159 156 176 175 126 at random from each of three different distributors Control: of mixed nuts and there are, respectively, 345, 313,

151 132 196 195 188 198 187 168 115 and 359 peanuts in each of the cans, can we conclude

165 137 208 133 217 191 193 140 146 at the 0.01 level of significance that the mixed nuts Can we conclude, at the 5% level of significance, that

of the three distributors contain equal proportions of the average cholesterol level has been reduced due to peanuts?

the program? Make the appropriate test on means. 10.104 A study was made to determine whether there

is a difference between the proportions of parents in 10.107 In a study conducted by the Department of the states of Maryland (MD), Virginia (VA), Georgia Mechanical Engineering and analyzed by the Statistics (GA), and Alabama (AL) who favor placing Bibles in Consulting Center at Virginia Tech, steel rods supplied the elementary schools. The responses of 100 parents by two different companies were compared. Ten sam- selected at random in each of these states are recorded ple springs were made out of the steel rods supplied by in the following table:

each company, and the “bounciness” was studied. The

State

data are as follows:

Preference MD VA GA AL

Company A:

Yes 65 71 78 82 9.3 8.8 6.8 8.7 8.5 6.7 8.0 6.5 9.2 7.0 No

35 29 22 18 Company B:

Can we conclude that the proportions of parents who 11.0 9.8 9.9 10.2 10.1 9.7 11.0 11.1 10.2 9.6 favor placing Bibles in the schools are the same for Can you conclude that there is virtually no difference these four states? Use a 0.01 level of significance.

in means between the steel rods supplied by the two 10.105 A study was conducted at the Virginia- companies? Use a P -value to reach your conclusion. Maryland Regional College of Veterinary Medicine Should variances be pooled here? Equine Center to determine if the performance of a certain type of surgery on young horses had any effect 10.108 In a study conducted by the Water Resources on certain kinds of blood cell types in the animal. Fluid Center and analyzed by the Statistics Consulting Cen- samples were taken from each of six foals before and af- ter at Virginia Tech, two different wastewater treat- ter surgery. The samples were analyzed for the number ment plants are compared. Plant A is located where of postoperative white blood cell (WBC) leukocytes. the median household income is below $22,000 a year,

A preoperative measure of WBC leukocytes was also and plant B is located where the median household measured. The data are given as follows:

income is above $60,000 a year. The amount of waste- Foal

Presurgery*

Postsurgery*

water treated at each plant (thousands of gallons/day) was randomly sampled for 10 days. The data are as

1 10.80 10.60 follows: 2 12.90 16.60 3 9.59 17.20 Plant A:

5 12.00 10.60 6 6.07 8.60 Plant B:

*All values × 10 −3 . 20 39 24 33 30 28 30 22 33 24 Use a paired sample t-test to determine if there is a sig- Can we conclude, at the 5% level of significance, that

386 Chapter 10 One- and Two-Sample Tests of Hypotheses the average amount of wastewater treated at the plant breast cancer reveals an average PCB concentration

in the high-income neighborhood is more than that of 22.8 × 10 −4 gram, with a standard deviation of treated at the plant in the low-income area? Assume

4.8 × 10 −4 gram, is the mean concentration of PCBs normality.

less than 24 × 10 −4 gram?

10.109 The following data show the numbers of de- 10.111 z-Value for Testing p 1 −p 2 =d 0 : To test fects in 100,000 lines of code in a particular type of the null hypothesis H 0 that p 1 −p 2 =d 0 , where d 0 software program developed in the United States and we base our decision on Japan. Is there enough evidence to claim that there is a significant difference between the programs developed

−d 0 in the two countries? Test on means. Should variances

p ˆ 1 −ˆ p 2

, p ˆ 1 q ˆ 1 /n 1 +ˆ p 2 q ˆ 2 /n 2 be pooled?

z=

U.S. 48 39 42 52 40 48 52 52 which is a value of a random variable whose distribu- 54 48 52 55 43 46 48 52

tion approximates the standard normal distribution as Japan

50 48 42 40 43 48 50 46 long as n 1 and n 2 are both large. With reference to 38 38 36 40 40 48 48 45

Example 10.11 on page 364, test the hypothesis that the percentage of town voters favoring the construction

10.110 Studies show that the concentration of PCBs of the chemical plant will not exceed the percentage of is much higher in malignant breast tissue than in county voters by more than 3%. Use a P -value in your normal breast tissue. If a study of 50 women with conclusion.