10.55 Consider an experiment with four groups, with eight values in each. For the ANOVA summary table below, fill in all the missing results: Source Degrees of Freedom Sum of Squares
Mean Square (Variance) F Among groups c – 1 = ? SSA = ? MSA = 80 FSTAT = ? Within groups n – c = ? SSW = 560 MSW = ? Total n – 1 = ? SST = ? 10.57 The Computer Anxiety Rating Scale (CARS) measures an individual’s level of computer anxiety, on a scale from
20 (no anxiety) to 100 (highest level of anxiety). Researchers at Miami University administered CARS to 172 business students. One of the objectives of the study was to determine whether there are differences in the amount of a. Complete the ANOVA summary
table. b. At the 0.05 level of significance, is there evidence of a difference in the mean computer anxiety experienced by different majors? c. If the results in (b) indicate that it is appropriate, use the Tukey-Kramer procedure to determine which majors
differ in mean computer anxiety. Discuss your findings. 10.59 A hospital conducted a study of the waiting time in its emergency room. The hospital has a main campus and three satellite locations. Management had a business objective of reducing waiting time
for emergency room cases that did not require immediate attention. To study this, a random sample of 15 emergency room cases that did not require immediate attention at each location were selected on a particular day, and the waiting time (measured from check-in
to when the patient was called into the clinic area) was measured. The results are stored in ERWaiting a. At the 0.05 level of significance, is there evidence of a difference in the mean waiting times in the four locations? b. If appropriate, determine which
locations differ in mean waiting time. c. At the 0.05 level of significance, is there evidence of a difference in the variation in waiting time among the four locations? 10.61 The per-store daily customer count (i.e., the mean number of customers in a store
in one day) for a nationwide convenience store chain that operates nearly 10,000 stores has been steady, at 900, for some time. To increase the customer count, the chain is considering cutting prices for coffee beverages. The question to be determined is how
much to cut prices to increase the daily customer count without reducing the gross margin on coffee sales too much. You decide to carry out an experiment in a sample of 24 stores where customer counts have been running almost exactly at the national average
of 900. In 6 of the stores, the price of a small coffee will now be $0.59, in 6 stores the price of a small coffee will now be $0.69, in 6 stores, the price of a small coffee will now be $0.79, and in 6 stores, the price of a small coffee will now be $0.89.
After four weeks of selling the coffee at the new price, the daily customer count in the stores was recorded and stored in . coffee sales a. At the 0.05 level of significance, is there evidence of a difference in the daily customer count based on the price
of a small coffee? b. If appropriate, determine which prices differ in daily customer counts. c. At the 0.05 level of significance, is there evidence of a difference in the variation in daily customer count among the different prices? d. What effect does your
result in (c) have on the validity of the results in (a) and (b)? 11.25 Where people turn for news is different for various age groups. A study indicated where different age groups primarily get their news: AGE GROUP MEDIA Under 36 36–50 50_ Local TV 107 119
133 National TV 73 102 127 Radio 75 97 109 Local newspaper 52 79 107 Internet 95 83 76 At the 0.05 level of significance, is there evidence of a significant relationship between the age group and where people primarily get their news? If so, explain the relationship.
Degrees of
Freedom
Sum of
Squares
Mean
Squares F
Among majors 5 3,172
Within majors 166 21,246
Total 171 24,418
Major n Mean
Marketing 19 44.37
Management 11 43.18
Other 14 42.21
Finance 45 41.80
Accountancy 36 37.56
MIS 47 32.21
Source: Data Extracted from T. Broome and D. Havelka, “Determinants
of Computer Anxiety in Business Students,” The Review of Business
Information Systems, Spring 2002, 6(2), pp. 9–16
Degrees of
Freedom
Sum of
Squares
Mean
Squares F p-value
Among chains 4 38,191.9096 9,547.9774 73.1086 0.0000
Within chains 95 12,407.00 130.60
Using the absolute differences given in Table 10.10, you perform a one-way ANOVA (see
Figure 10.21).
From the Figure 10.21 Excel results, observe that (Excel labels this
value F. Minitab labels the value Test statistic and reports a value of 0.21.) Because
(or the ), you do not reject There is
no evidence of a significant difference among the four variances. In other words, it is reasonable
to assume that the materials from the four suppliers produce parachutes with an
equal amount of variability. Therefore, the homogeneity-of-variance assumption for the
ANOVA procedure is justified.
Example 10.5 illustrates another example of the one way ANOVA.
FSTAT = 0.2068 6 3.2389 p-value = 0.8902 7 0.05 H0.
FSTAT = 0.2068.
At the 0.05 level of significance, is there evidence of a difference in the mean drivethrough
service times of the five chains?
The Tukey-Kramer Procedure
In the parachute company example, you used the one-way ANOVA F test to determine that
there was a difference among the suppliers. The next step is to construct multiple comparisons
to determine which suppliers are different.
Although many procedures are available (see references 3 and 4), this text uses the
Tukey-Kramer multiple comparisons procedure for one-way ANOVA to determine
which of the c means are significantly different. The Tukey-Kramer procedure enables you
a
FSTAT
F I G U R E 1 0 . 1 9
Excel and Minitab ANOVA results for the parachute experiment
Figure 10.19 shows the ANOVA results for the parachute experiment, including the
p-value. In Figure 10.19, what Table 10.7 (see page 369) labels Among Groups is labeled
Between Groups in the Excel results and Factor in the Minitab results. What Table 10.8 labels
Within Groups is labeled Error in the Minitab results.
ISBN 1-269-14496-0
Business Statistics: A First Course, Sixth Edition, by David M. Levine, Timothy C. Krehbiel, and Mark L. Berenson. Published by Prentice Hall.
Copyright © 2013 by Pearson Education, Inc.
10.5 One-Way Analysis of Variance 373
to simultaneously make comparisons between all pairs of groups. You use the following
four steps to construct the comparisons:
1. Compute the absolute mean differences, (where ), among all
pairs of sample means.
2. Compute the critical range for the Tukey-Kramer procedure, using Equation (10.13).
ƒ Xj – Xj¿ ƒ j Z j¿ c1c – 12>2
CRITICAL RANGE FOR THE TUKEY-KRAMER PROCEDURE
(10.13)
wwhere is the upper-tail critical value from a Studentized range distribution having c
degrees of freedom in the numerator and degrees of freedom in the denominator.
(Values for the Studentized range distribution are found in Table E.6.)
n – c
Qa
Critical range = QaA
MSW
2
a 1
nj
+
1
nj¿
b
If the sample sizes differ, you compute a critical range for each pairwise comparison of sample
means.
3. Compare each of the pairs of means against its corresponding critical range.
You declare a specific pair significantly different if the absolute difference in the sample
means, is greater than the critical range.
4. Interpret the results.
In the parachute example, there are four suppliers. Thus, there are y 1 = 2 = 5x
y intercept b o
x =3
2.5 Zagat’s publishes restaurant ratings for various locations
in the United States. The file Restaurants contains the
Zagat rating for food, decor, service, and the cost per person
for a sample of 100 restaurants located in New York City and
in a suburb of New York City. Develop a regression model to
predict the price per person, based on a variable that represents
the sum of the ratings for food, decor, and service.
Sources: Extracted from Zagat Survey 2010, New York City
Restaurants; and Zagat Survey 2009–2010, Long Island Restaurants.
a. Construct a scatter plot.
For these data, b o = -28.1975 and b1 = 1.2409
b. Assuming a linear cost relationship, use the least-squares
method to compute the regression coefficients and
c. Interpret the meaning of the Y intercept, and the
slope, in this problem.
d. Predict the cost per person for a restaurant with a summated
rating of 50.
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