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ECO252 QBA2
THIRD HOUR EXAM
April 16, 2007
Name
Student number_____________
I. (8 points) Do all the following (2points each unless noted otherwise). Make Diagrams! Show your
work!
x ~ N 13, 7.2
1. Px  0
2. P6  x  20 
3. P1  x  12 
4. x.145 (Find z .145 first)
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II. II. (22+ points) Do all the following (2 points each unless noted otherwise). Do not answer a question
‘yes’ or ‘no’ without giving reasons. Show your work when appropriate. Use a 5% significance level except
where indicated otherwise.
1. Turn in your computer problems 2 and 3 marked as requested in the Take-home. (5 points, 2 point
penalty for not doing.)
2. (Dummeldinger) As part of a study to investigate the effect of helmet design on football injuries,
head width measurements were taken for 30 subjects randomly selected from each of 3 groups (High
school football players, college football players and college students who do not play football – so that
there are a total of 90 observations) with the object of comparing the typical head widths of the three
groups. Before the data was compared, researchers ran two tests on it. They first ran a Lilliefors test
and found that the null hypothesis was not rejected. They then ran a Bartlett test and found that the null
hypothesis was not rejected. On the basis of these results they should use the following method.
a) The Kruskal-Wallis test.
b) One-way ANOVA
c) The Friedman test
d) Two-Way ANOVA
e) Chi-square test
[7]
3. The coefficient of determination
a) Is the square of the sample correlation
b) Cannot be negative
c) Gives the fraction of the variation in the dependent variable explained by the
independent variable.
d) All of the above.
e) None of the above.
4. The Kruskal-Wallis test is an extension of which of the following for two independent samples?
a) Pooled-variance t test
b) Paired-sample t test
c) (Mann – Whitney -) Wilcoxon rank sum test
d) Wilcoxon signed ranks test
e) A test for comparison of means for samples with unequal variances
f) None of the above.
5. One-way ANOVA is an extension of which of the following for two independent samples?
a) Pooled-variance t test
b) Paired-sample t test
c) (Mann – Whitney -) Wilcoxon rank sum test
d) Wilcoxon signed ranks test
e) A test for comparison of means for samples with unequal variances
f) None of the above.
[13]
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Exhibit 1: As part of an evaluation program, a sporting goods retailer wanted to compare the downhill
coasting speeds of 4 brands of bicycles. She took 3 of each brand and determined their maximum
downhill speeds. The results are presented in miles per hour in the table below. She assumes that the
Normal distribution is not applicable and uses a rank test. She treats the columns as independent
random samples.
Barth Tornado Reiser Shaw
43
37
41
43
46
38
45
45
43
39
42
46
6. What are the null and alternative hypotheses?
7. Construct a table of ranks for the data in Exhibit 1.
8. Complete the test using a 5% significance level (3)
[20]
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Exhibit 2: (Webster) A placement director is trying to find out if a student’s GPA can influence how
many job offers the student receives. Data is assembled and run on Minitab, with the results below.
The regression equation is
Jobs = - 0.248 + 1.27 GPA
Predictor
Coef SE Coef
Constant
-0.2481
0.7591
GPA
1.2716
0.2859
S = 1.03225
R-Sq = 71.2%
Analysis of Variance
Source
DF
SS
Regression
1 21.076
Residual Error
8
8.524
Total
9 29.600
T
P
-0.33 0.752
4.45 0.002
R-Sq(adj) = 67.6%
MS
21.076
1.066
F
19.78
P
0.002
9. In Exhibit 2, how many job offers would you predict for someone with a GPA of 3.
10. In Exhibit 2, what percent of the variation in job offers is explained by variation of GPA? (1)
11. For the regression in Exhibit 2 to give us ‘good’ estimates of the coefficients.
a) The variances of ‘Jobs’ and ‘GPA’ must be equal.
b) The amount of variation of ‘Jobs’ around the regression line must be independent of ‘GPA.’
c) ‘Jobs’ must be independent of ‘GPA.’
d) The p-value for ‘Constant’ must be higher than the p-value for ‘GPA.’
e) All of the above must be true.
f) None of the above need to be true.
[25]
12. Assuming that the Total of 29.600 in the ANOVA in Exhibit 2 is correct, but that you cannot read any
other part of the printout, what are the largest and smallest values that the Regression Sum of Squares could
take?
[27]
13. A corporation wishes to test the effectiveness of 3 different designs of pollution-control devices. It takes
its 6 factories and puts each type of device for one week during each month of the year, for a total of 216
measurements. Pollution in the smoke is measured and an ANOVA is begun. Let factor A be months. Factor
B will be factories and Factor C will be the devices. Fill in the following table. Note that we have too little
data to compute interaction ABC. The Total sum of squares is 106. SSA is 44, SSB is 11, SSC is 21, SSAB
is 9, SSAC is 7 and SSBC is 3. Use a significance level of 1%. Do we have enough information to decide
which of the devices is best? (Do not answer this without showing your work!) (6) [33]
Row
1
2
3
4
5
6
7
8
Factor
A
B
C
AB
AC
BC
Within
Total
SS
DF
MS
F
F01
Significant?
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Exhibit 3: A plant manager is concerned for the rising blood pressure of employees and believes that it is
related to the level of satisfaction that they get out of their jobs. A random sample of 10 employees rates
their satisfaction on a 1to 60 scale and their blood pressure is tested.
Row
1
2
3
4
5
6
7
8
9
10
Satisfaction BloodPr
34
124
23
128
19
157
43
133
56
116
47
125
32
147
16
167
55
110
25
156
We also know that Sum of Satisfaction = 350, Sum of BloodPr = 1363, Sum of squares (uncorrected) of
Satisfaction = 14170 and Sum of squares (uncorrected) of BloodPr = 189113. It is also true that the sum of
the first eight numbers of the product of Satisfaction and Blood is 35609.
14. a) Finish the computation of
 xy (1). Try to do a regression explaining blood pressure as a result of
job satisfaction (4). Please do not recompute things that I have already done for you. Find the slope and test
its significance(4). Do the prediction or confidence interval as appropriate for the blood pressure of the
highly satisfied employee (satisfaction level is 57) that you hired last month. (3)
[45]
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ECO252 QBA2
THIRD EXAM
April 16, 2007
TAKE HOME SECTION
Name: _________________________
Student Number: _________________________
Class days and time : _________________________
Please Note: Computer problems 2 and 3 should be turned in with the exam (2). In problem 2, the 2 way
ANOVA table should be checked. The three F tests should be done with a 5% significance level and you
should note whether there was (i) a significant difference between drivers, (ii) a significant difference
between cars and (iii) significant interaction. In problem 3, you should show on your third graph where the
regression line is. Check what your text says about normal probability plots and analyze the plot you did.
Explain the results of the t and F tests using a 5% significance level. (2)
III Do the following. (20+ points) Note: Look at 252thngs (252thngs) on the syllabus supplement part of
the website before you start (and before you take exams). Show your work! State H 0 and H 1 where
appropriate. You have not done a hypothesis test unless you have stated your hypotheses, run the
numbers and stated your conclusion. (Use a 95% confidence level unless another level is specified.)
Answers without reasons or accompanying calculations usually are not acceptable. Neatness and
clarity of explanation are expected. This must be turned in when you take the in-class exam. Note
that from now on neatness means paper neatly trimmed on the left side if it has been torn, multiple
pages stapled and paper written on only one side.
1) (Ken Black) A national travel organization wanted to determine whether there is a significant difference
between the cost of premium gas in various regions of the country. The following data was gathered in July
2006. Because the brand of gas may confuse the results, data is blocked by brand.
Data Display
Row
1
2
3
4
5
6
Brand
A
B
C
D
E
F
City 1
3.47
3.43
3.44
3.46
3.46
3.44
City 2
3.40
3.41
3.41
3.45
3.40
3.43
City 3
3.38
3.42
3.43
3.40
3.39
3.42
City 4
3.32
3.35
3.36
3.30
3.39
3.39
City 5
3.50
3.44
3.45
3.45
3.48
3.49
To personalize these data, let the last digit of your student number be a . Mark your problem ‘VERSION
a .’ If a is zero, use the original numbers. If a is 1 through 4, add a
to the city 4 column. If a is 5
100
through 9, add 0.04  a
to the city 5 column. Example: Seymour Butz’ number is 090909, so he adds
100
0.04  9
 .05 to column 5 (subtracts .05) and gets the table below.
100
Version 9
Row
1
2
3
4
5
6
Brand
A
B
C
D
E
F
City 1
3.47
3.43
3.44
3.46
3.46
3.44
City 2
3.40
3.41
3.41
3.45
3.40
3.43
City 3
3.38
3.42
3.43
3.40
3.39
3.42
City 4
3.32
3.35
3.36
3.30
3.39
3.39
City 5
3.45
3.39
3.40
3.40
3.43
3.44
a) Do a 2-way ANOVA on these data and explain what hypotheses you test and what the conclusions are.
Show your work. (6)
b) Using your results from a) present two different confidence intervals for the difference between numbers
of defects for the best and worst worker and for the defects from the best and second best times. Explain
which of the intervals show a significant difference and why. (3)
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c) What other method could we use on these data to see if city makes a difference while allowing for crossclassification? Under what circumstances would we use it? Try it and tell what it tests and what it shows. (3)
d) (Extra credit) Check you results on the computer. The Minitab setup for both 2-way ANOVA and the
corresponding non-parametric method is the same as the extra credit in the third graded assignment or the
third computer problem. You may want to use and compare the Statistics pull-down menu and ANOVA or
Nonparametrics to start with. Explain your results. (4) [12]
2) A local official wants to know how land value affects home price. The official collects the following
data. 'Home price' is the dependent variable and 'Land Value' is the independent variable. Both are in
thousands of dollars. If you don’t know what dependent and independent variables are, stop work
until you find out.
Row
1
2
3
4
5
6
7
8
9
10
HomePr
67
63
60
54
58
36
76
87
89
92
LandVal
7.0
6.9
5.5
3.7
5.9
3.8
8.9
9.6
9.9
10.0
To personalize the data let g be the second to last digit in your student number. Mark your problem
‘VERSION g . ’Subtract g from 92, which is the last home price.
a) Compute the regression equation Y  b  b x to predict the home price on the basis of land value.
0
1
(3).You may check your results on the computer, but let me see real or simulated hand calculations.
b) Compute R 2 . (2)
c) Compute s e . (2)
d) Compute s b0 and do a significance test on b0 (2)
e) Use your equation to predict the price of the average home that is on land worth $9000. Make this into a
95% interval. (3)
f) Make a graph of the data. Show the trend line and the data points clearly. If you are not willing to do this
neatly and accurately, don’t bother. (2)
[26]
3) A sales manager is trying to find out what method of payment works best. Salespersons are randomly
assigned to payment by commission, salary or bonus. Data for units sold in a week is below.
Row
1
2
3
4
5
6
7
8
9
10
11
12
13
Commission
25
35
20
30
25
30
26
17
31
30
Salary
25
25
22
20
25
23
22
25
20
26
24
Bonus
28
41
21
31
26
26
46
32
27
35
23
22
26
To personalize the data let g be the second to last digit in your student number. Subtract 2 g from 46 in
the last column. Mark your problem ‘VERSION g .’Example: Payme Well’s student number is 000090, so
she subtracts 18 and gets the following numbers.
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Version 9
Row
1
2
3
4
5
6
7
8
9
10
11
12
13
Commission
25
35
20
30
25
30
26
17
31
30
Salary
25
25
22
20
25
23
22
25
20
26
24
Bonus
28
41
21
31
26
26
28
32
27
35
23
22
26
a) State your null hypothesis and test it by doing a 1-way ANOVA on these data and explain whether the
test tells us that payment method matters or not. (4)
b) Using your results from a) present individual and Tukey intervals for the difference between earnings for
all three possible contrasts. Explain (i) under what circumstances you would use each interval and (ii)
whether the intervals show a significant difference. (2)
c) What other method could we use on these data to see if payment method makes a difference? Under what
circumstances would we use it? Try it and tell what it tests and what it shows. (3) [37]
d) (Extra Credit) Do a Levene test on these data and explain what it tests and shows. (4)
f) (Extra credit) Check you results on the computer. The Minitab setup for 1-way ANOVA can either be
unstacked (in columns) or stacked (one column for data and one for column name). There is a Tukey
option. The corresponding non-parametric methods require the stacked presentation. You may want to use
and compare the Statistics pull-down menu and ANOVA or Nonparametrics to start with. You may also
want to try the Mood median test and the test for equal variances which are on the menu categories above.
Even better, do a normality test on your columns. Use the Statistics pull-down menu to find the normality
tests. The Kolmogorov-Smirnov option is actually Lilliefors. The ANOVA menu can check for equality of
variances. In light of these tests was the 1-way ANOVA appropriate? You can get descriptions of unfamiliar
tests by using the Help menu and the alphabetic command list or the Stat guide. (Up to 7) [12]
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