Probability and statistics for engineers and scientists - Ross

470 Chapter 10: Analysis of Variance

The p-values in a Two-way ANOVA with a Possible Interaction

Enter the number of rows: 3 Begin Data Entry
Quit
Enter the number of columns: 2

Enter the number of 4
observations in each cell:

FIGURE 10.3

The p-values in a Two-way ANOVA with Possible Interaction

Click on a cell to enter data Start
Clear All Observations
AB
1 135, 150, 176, 85 50, 55, 64, 38
2 150, 162, 171, 120 76, 88, 91, 57
3 138, 111, 140, 106 68, 60, 74, 51

The value of the F-statistic for testing that there is no row effect is 2.47976
The p-value for testing that there is no row effect is 0.1093
The value of the F-statistic for testing that there is no column effect is 69.63223
The p-value for testing that there is no column effect is less than 0.0001
The value of the F-statistic for testing that there is no interaction effect is 0.64625
The p-value for testing that there is no interaction effect is 0.5329

FIGURE 10.4

Problems 471

Problems

1. A purification process for a chemical involves passing it, in solution, through a
resin on which impurities are adsorbed. A chemical engineer wishing to test the
efficiency of 3 different resins took a chemical solution and broke it into 15 batches.
She tested each resin 5 times and then measured the concentration of impurities
after passing through the resins. Her data were as follows:

Concentration of Impurities

Resin I Resin II Resin III

.046 .038 .031
.025 .035 .042
.014 .031 .020
.017 .022 .018
.043 .012 .039

Test the hypothesis that there is no difference in the efficiency of the resins.

2. We want to know what type of filter should be used over the screen of a cathode-
ray oscilloscope in order to have a radar operator easily pick out targets on the
presentation. A test to accomplish this has been set up. A noise is first applied to
the scope to make it difficult to pick out a target. A second signal, representing the
target, is put into the scope, and its intensity is increased from zero until detected
by the observer. The intensity setting at which the observer first notices the target
signal is then recorded. This experiment is repeated 20 times with each filter. The
numerical value of each reading listed in the table of data is proportional to the
target intensity at the time the operator first detects the target.

Filter No. 1 Filter No. 2 Filter No. 3

90 88 95
87 90 95
93 97 89
96 87 98
94 90 96
88 96 81
90 90 92
84 90 79
101 100 105
96 93 98
90 95 92

(continued )

472 Chapter 10: Analysis of Variance

Filter No. 1 Filter No. 2 Filter No. 3

82 86 85
93 89 97
90 92 90
96 98 87
87 95 90
99 102 101
101 105 100
79 85 84
98 97 102

Test, at the 5 percent level of significance, the hypothesis that the filters are the
same.

3. Explain why we cannot efficiently test the hypothesis H0 : µ1 = µ2 = · · · = µm
m
by running t-tests on all of the 2 pairs of samples.

4. A machine shop contains 3 ovens that are used to heat metal specimens. Subject to

random fluctuations, they are all supposed to heat to the same temperature. To test

this hypothesis, temperatures were noted on 15 separate heatings. The following

data resulted.

Oven Temperature

1 492.4, 493.6, 498.5, 488.6, 494
2 488.5, 485.3, 482, 479.4, 478
3 502.1, 492, 497.5, 495.3, 486.7

Do the ovens appear to operate at the same temperature? Test at the 5 percent
level of significance. What is the p-value?

5. Four standard chemical procedures are used to determine the magnesium content
in a certain chemical compound. Each procedure is used four times on a given
compound with the following data resulting.

Method

1 23 4

76.42 80.41 74.20 86.20
78.62 82.26 72.68 86.04
80.40 81.15 78.84 84.36
78.20 79.20 80.32 80.68

Do the data indicate that the procedures yield equivalent results?

Problems 473

6. Twenty overweight individuals, each more than 40 pounds overweight, were ran-
domly assigned to one of two diets. After 10 weeks, the total weight losses (in
pounds) of the individuals on each of the diets were as follows:

Weight Loss

Diet 1 Diet 2

22.2 24.2
23.4 16.8
24.2 14.6
16.1 13.7
9.4 19.5
12.5 17.6
18.6 11.2
32.2 9.5

8.8 30.1
7.6 21.5

Test, at the 5 percent level of significance, the hypothesis that the two diets have
equal effect.

7. In a test of the ability of a certain polymer to remove toxic wastes from water, exper-
iments were conducted at three different temperatures. The data below give the
percentages of the impurities that were removed by the polymer in 21 independent
attempts.

Low Temperature Medium Temperature High Temperature

42 36 33
41 35 44
37 32 40
29 38 36
35 39 44
40 42 37
32 34 45

Test the hypothesis that the polymer performs equally well at all three temperatures.
Use the (a) 5 percent level of significance and (b) 1 percent level of significance.

8. In the one-factor analysis of variance model with n observations per sample, let Si2,
i = 1, . . . , m denote the sample variances for the m samples. Show that

m

SSW = (n − 1) Si2

i=1

474 Chapter 10: Analysis of Variance

9. The following data relate to the ages at death of a certain species of rats that were
fed 1 of 3 types of diet. Thirty rats of a type having a short life span were randomly
divided into 3 groups of size 10 each. The sample means and sample variances of
the ages at death (measured in months) of the 3 groups are as follows:

Sample mean Very Low Calorie Moderate Calorie High Calorie
Sample variance
22.4 16.8 13.7
24.0 23.2 17.1

Test the hypothesis, at the 5 percent level of significance, that the mean lifetime
of a rat is not affected by its diet. What about at the 1 percent level?

10. Plasma bradykininogen levels are related to the body’s ability to resist inflamma-
tion. In a 1968 study (Eilam, N., Johnson, P. K., Johnson, N. L., and Creger, W.,
“Bradykininogen levels in Hodgkin’s disease,” Cancer, 22, pp. 631–634), levels
were measured in normal patients, in patients with active Hodgkin’s disease, and
in patients with inactive Hodgkin’s disease. The following data (in micrograms of
bradykininogen per milliliter of plasma) resulted.

Normal Active Hodgkin’s Disease Inactive Hodgkin’s Disease

5.37 3.96 5.37
5.80 3.04 10.60
4.70 5.28
5.70 3.40 5.02
3.40 4.10 14.30
8.60 3.61
7.48 6.16 9.90
5.77 3.22 4.27
7.15 7.48 5.75
6.49 3.87 5.03
4.09 4.27 5.74
5.94 4.05 7.85
6.38 2.40 6.82
7.90
8.36

Test, at the 5 percent level of significance, the hypothesis that the mean
bradykininogen levels are the same for all three groups.

11. A study of the trunk flexor muscle strength of 75 girls aged 3 to 7 was reported
by Baldauf, K., Swenson, D., Medeiros, J., and Radtka, S., “Clinical assessment
of trunk flexor muscle strength in healthy girls 3 to 7,” Physical Therapy, 64, pp.
1203–1208, 1984. With muscle strength graded on a scale of 0 to 5, and with 15
girls in each age group, the following sample means and sample standard deviations
resulted.

Problems 475

Age 3 4 5 6 7

Sample mean 3.3 3.7 4.1 4.4 4.8

Sample standard deviation .9 1.1 1.1 .9 .5

Test, at the 5 percent level of significance, the hypothesis that the mean trunk
flexor strength is the same for all five age groups.

12. An emergency room physician wanted to know whether there were any differences
in the amount of time it takes for three different inhaled steroids to clear a mild
asthmatic attack. Over a period of weeks she randomly administered these steroids
to asthma sufferers, and noted the time it took for the patients’ lungs to become
clear. Afterward, she discovered that 12 patients had been treated with each type
of steroid, with the following sample means (in minutes) and sample variances
resulting.

Steroid Xi Si2

A 32 145
B 40 138
C 30 150

(a) Test the hypothesis that the mean time to clear a mild asthmatic attack is the
same for all three steroids. Use the 5 percent level of significance.

(b) Find confidence intervals for all quantities µi − µj that, with 95 percent
confidence, are valid.

13. Five servings each of three different brands of processed meat were tested for fat
content. The following data (in fat percentage per gram) resulted.

Brand 1 2 3

32 41 36
Fat 34 32 37
content 31 33 30

35 29 28
33 35 33

(a) Does the fat content differ depending on the brand?
(b) Find confidence intervals for all quantities µi − µj that, with 95 percent

confidence, are valid.

14. A nutritionist randomly divided 15 bicyclists into 3 groups of 5 each. The first
group was given a vitamin supplement to take with each of their meals during the

476 Chapter 10: Analysis of Variance

next 3 weeks. The second group was instructed to eat a particular type of high-fiber
whole-grain cereal for the next 3 weeks. The final group was instructed to eat as
they normally do. After the 3-week period elapsed, the nutritionist had each of the
bicyclists ride 6 miles. The following times were recorded.

Vitamin group: 15.6 16.4 17.2 15.5 16.3
Fiber cereal group: 17.1 16.3 15.8 16.4 16.0
Control group: 15.9 17.2 16.4 15.4 16.8

(a) Are the data consistent with the hypothesis that neither the vitamin nor
the fiber cereal affected the bicyclists’ speeds? Use the 5 percent level of
significance.

(b) Find confidence intervals for all quantities µi − µj that, with 95 percent
confidence, are valid.

15. Test the hypothesis that the following three independent samples all come from
the same normal probability distribution.

Sample 1 Sample 2 Sample 3

35 29 44
37 38 52
29 34 56
27 30
30 32

16. For data xij , i = 1, . . . , m, j = 1, . . . , n, show that

mn

x.. = xi./m = x.j /n

i=1 j=1

17. If xij = i + j2, determine

32

(a) j=1 i=1xij

23

(b) i=1 j=1xij
18. If xij = ai + bj , show that

mn m n

xij = n ai + m bj

i=1 j=1 i=1 j=1

19. A study has been made on pyrethrum flowers to determine the content of pyrethrin,
a chemical used in insecticides. Four methods of extracting the chemical are used,

Problems 477

and samples are obtained from flowers stored under three conditions: fresh flowers,
flowers stored for 1 year, and flowers stored for 1 year but treated. It is assumed
that there is no interaction present. The data are as follows:

Pyrethrin Content, Percent

Storage A Method D
Condition BC

1 1.35 1.13 1.06 .98
2 1.40 1.23 1.26 1.22
3 1.49 1.46 1.40 1.35

Suggest a model for the preceding information, and use the data to estimate its
parameters.

20. The following data refer to the number of deaths per 10,000 adults in a large
Eastern city in the different seasons for the years 1982 to 1986.

Year Winter Spring Summer Fall

1982 33.6 31.4 29.8 32.1
1983 32.5 30.1 28.5 29.9
1984 35.3 33.2 29.5 28.7
1985 34.4 28.6 33.9 30.1
1986 37.3 34.1 28.5 29.4

(a) Assuming a two-factor model, estimate the parameters.
(b) Test the hypothesis that death rates do not depend on the season. Use the

5 percent level of significance.
(c) Test, at the 5 percent level of significance, the hypothesis that there is no

effect due to the year.

21. For the model of Problem 19:

(a) Do the methods of extraction appear to differ?
(b) Do the storage conditions affect the content? Test at the α = .05 level of

significance.

22. Three different washing machines were employed to test four different detergents.
The following data give a coded score of the effectiveness of each washing.

(a) Estimate the improvement in mean value when using detergent 1 over using
detergents (i) 2; (ii) 3; (iii) 4.

(b) Estimate the improvement in mean value when using machine 3 as opposed
to using machine (i) 1; (ii) 2.

478 Chapter 10: Analysis of Variance

Detergent 1 Machine 3
Detergent 2
Detergent 3 12 59
Detergent 4 60
53 50 62
54 54 57
56 58
50 45

(c) Test the hypothesis that the detergent used does not affect the score.
(d) Test the hypothesis that the machine used does not affect the score.

Use, in both (c) and (d), the 5 percent level of significance.

23. An experiment was devised to test the effects of running 3 different types of gasoline
with 3 possible types of additive. The experiment called for 9 identical motors to
be run with 5 gallons for each of the pairs of gasoline and additives. The following
data resulted.

Mileage Obtained

Additive

Gasoline 1 2 3

1 124.1 131.5 127

2 126.4 130.6 128.4

3 127.2 132.7 125.6

(a) Test the hypothesis that the gasoline used does not affect the mileage.
(b) Test the hypothesis that the additives are equivalent.
(c) What assumptions are you making?

24. Suppose in Problem 6 that the 10 people placed on each diet consisted of 5 men
and 5 women, with the following data.

Diet 1 Diet 2

Women 7.6 19.5
8.8 17.6
12.5 16.8
16.1 13.7
18.6 21.5

Men 22.2 30.1
23.4 24.2
24.2 9.5
32.2 14.6
9.4 11.2

Problems 479

(a) Test the hypothesis that there is no interaction between gender and diet.
(b) Test the hypothesis that the diet has the same effect on men and women.

25. A researcher is interested in comparing the breaking strength of different laminated
beams made from 3 different types of glue and 3 varieties of wood. To make the
comparison, 5 beams of each of the 9 combinations were manufactured and then
put under a stress test. The following table indicates the pressure readings at which
each of the beams broke.

lGlue G1 G2 G3

Wood 196 208 214 216 258 250
247 216 235 240 264 248
W1 221 252 272
W2 216 228 215 217 246 247
W3 240 224 235 219 261 250
236 241 255
230 242 212 218 255 251
232 244 216 224 261 258
228 222 247

(a) Test the hypothesis that the wood and glue effect is additive.
(b) Test the hypothesis that the wood used does not affect the breaking strength.
(c) Test the hypothesis that the glue used does not affect the breaking strength.

26. A study was made as to how the concentration of a certain drug in the blood,
24 hours after being injected, is influenced by age and gender. An analysis of the
blood samples of 40 people given the drug yielded the following concentrations
(in milligrams per cubic centimeter).

Male 11–25 Age Group Over 65
Female
52 26–40 41–65 82.4
56.6 86.2
68.2 52.5 53.2 101.3
82.5 49.6 53.6 92.4
85.6 48.7 49.8 78.6
68.6 44.6 50.0 82.2
80.4 43.4 51.2 79.6
86.2 60.2 58.7 81.4
81.3 58.4 55.9 80.6
77.2 56.2 56.0 82.2
54.2 57.2
61.1 60.0

480 Chapter 10: Analysis of Variance

(a) Test the hypothesis of no age and gender interaction.
(b) Test the hypothesis that gender does not affect the blood concentration.
(c) Test the hypothesis that age does not affect blood concentration.

27. Suppose, in Problem 23, that there has been some controversy about the assump-
tion of no interaction between gasoline and additive used. To allow for the
possibility of an interaction effect between gasoline and additive, it was decided to
run 36 motors — 4 in each grouping. The following data resulted.

Gasoline 1 Additive 3

1 126.2 2 127
124.8 126.6
2 125.3 130.4 129.4
127.0 131.6 130.1
3 127.2 132.5 129.5
126.6 128.6 142.6
125.8 142.1 140.5
128.4 132.6 138.7
127.1 128.5 125.2
128.3 131.2 123.3
125.1 132.3 122.6
124.9 134.1 120.9
130.6
133.0

(a) Do the data indicate an interaction effect?
(b) Do the gasolines appear to give equal results?
(c) Test whether or not there is an additive effect or whether all additives work

equally well.
(d) What conclusions can you draw?

28. An experiment has been devised to test the hypothesis that an elderly person’s
memory retention can be improved by a set of “oxygen treatments.” A group of
scientists administered these treatments to men and women. The men and women
were each randomly divided into 4 groups of 5 each, and the people in the ith
group were given treatments over an (i − 1) week interval, i = 1, 2, 3, 4. (The
2 groups not given any treatments served as “controls.”) The treatments were set
up in such a manner that all individuals thought they were receiving the oxygen
treatments for the total 3 weeks. After treatment ended, a memory retention
test was administered. The results (with higher scores indicating higher memory
retentions) are shown in the table.

(a) Test whether or not there is an interaction effect.
(b) Test the hypothesis that the length of treatment does not affect memory

retention.

Problems 481

(c) Is there a gender difference?
(d) A randomly chosen group of 5 elderly men, without receiving any oxygen

treatment, were given the memory retention test. Their scores were 37, 35,
33, 39, 29. What conclusions can you draw?

Scores Number of Weeks of Oxygen Treatment
Men
Women 012 3

42 39 38 42
54 52 50 55
46 51 47 39
38 50 45 38
51 47 43 51
49 48 27 61
44 51 42 55
50 52 47 45
45 54 53 40
43 40 58 42

29. In a study of platelet production, 16 rats were put at an altitude of 15,000 feet,
while another 16 were kept at sea level (Rand, K., Anderson, T., Lukis, G., and
Creger, W., “Effect of hypoxia on platelet level in the rat,” Clinical Research, 18,

Altitude Spleen Removed Normal Spleen
Sea Level
528 434
444 331
338 312
342 575
338 472
331 444
288 575
319 384
294 272
254 275
352 350
241 350
291 466
175 388
241 425
238 344

482 Chapter 10: Analysis of Variance

p. 178, 1970). Half of the rats in both groups had their spleens removed. The
fibrinogen levels on day 21 are reported below.
(a) Test the hypothesis that there are no interactions.
(b) Test the hypothesis that there is no effect due to altitude.
(c) Test the hypothesis that there is no effect due to spleen removal. In all cases,

use the 5 percent level of significance.
Suppose that µ, α1, . . . , αm, β1, . . . , βn and µ , α1, . . . , αm, β1, . . . , βn are such
that

µ + αi + βj = µ + αi + βj for all i, j
αi = αi = βj = βj = 0

iijj

Show that
µ = µ , αi = αi, βj = βj

for all i and j. This shows that the parameters µ, α1, . . . , αm, β1, . . . , βn in our
representation of two factor ANOVA are uniquely determined.

Chapter 11

GOODNESS OF FIT TESTS AND
CATEGORICAL DATA ANALYSIS

11.1 INTRODUCTION

We are often interested in determining whether or not a particular probabilistic model
is appropriate for a given random phenomenon. This determination often reduces to
testing whether a given random sample comes from some specified, or partially specified,
probability distribution. For example, we may a priori feel that the number of industrial
accidents occurring daily at a particular plant should constitute a random sample from
a Poisson distribution. This hypothesis can then be tested by observing the number of
accidents over a sequence of days and then testing whether it is reasonable to suppose
that the underlying distribution is Poisson. Statistical tests that determine whether a given
probabilistic mechanism is appropriate are called goodness of fit tests.

The classical approach to obtaining a goodness of fit test of a null hypothesis that
a sample has a specified probability distribution is to partition the possible values of the
random variables into a finite number of regions. The numbers of the sample values
that fall within each region are then determined and compared with the theoretical
expected numbers under the specified probability distribution, and when they are signifi-
cantly different the null hypothesis is rejected. The details of such a test are presented
in Section 11.2, where it is assumed that the null hypothesis probability distribution is
completely specified. In Section 11.3, we show how to do the analysis when some of the
parameters of the null hypothesis distribution are left unspecified; that is, for instance, the
null hypothesis might be that the sample distribution is a normal distribution, without
specifying the mean and variance of this distribution. In Sections 11.4 and 11.5, we con-
sider situations where each member of a population is classified according to two distinct
characteristics, and we show how to use our previous analysis to test the hypothesis that
the characteristics of a randomly chosen member of the population are independent. As
an application, we show how to test the hypothesis that m population all have the same
discrete probability distribution. Finally, in the optional section, Section 11.6, we return

483