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ECO252 QBA2
THIRD HOUR EXAM
Nov 30 and Dec 1, 2006
Name
Hour of Class Registered (Circle)
MWF 1 MWF 2 TR 12:30 TR 2
I. (8 points) Do all the following (2points each unless noted otherwise). Make Diagrams! Show your
work!
x ~ N 15, 9.3
1. Px  20 
2. P0  x  14 
3. P16  x  16 
4. x.42 (Find z .42 first)
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II. (22+ points) Do all the following (2points each unless noted otherwise). Do not answer question ‘yes’
or ‘no’ without giving reasons. Show your work in questions that are not multiple choice. Look them
over first. The exam is normed on 50 points.
Note the following:
1. This test is normed on 50 points, but there are more points possible including the take-home.
You are unlikely to finish the exam and might want to skip some questions.
2. If you answer ‘None of the above’ in any question, you should provide an alternative
answer and explain why. You may receive credit for this even if you are wrong.
3. Use a 5% significance level unless the question says otherwise.
4. Read problems carefully. A problem that looks like a problem on another exam may be
quite different.
1. Turn in your computer problems 2 and 3 marked to show the following: (5 points, 2 point penalty for
not doing.)
a) In computer problem 2 – what is tested and what are the results?
b) In computer problem 3 – what coefficients are significant? What is your evidence?
c) In the last graph in computer problem 3, where is the regression line?
[5]
2. (Abronovic) The distance that a baseball travels after being hit is a function of the velocity (in mph)
of the pitched ball. A ball is pitched to a batter with a 35 inch, 32 oz bat that is swung at 70mph from
the waist and at an angle of 35%. The experiment is repeated 9 times. A partial Minitab printout
appears below. Use   .01 throughout this problem.
DIST = ………
Predictor
Constant
VELOC
s = 1.185
+ ……… VELOC
Coef
StDev
t-ratio
p
31.311
0.999
……………
…………
0.74667 0.01529 ……………
…………
R-sq = ……… Rsq(adj) = ………
Analysis of Variance
SOURCE
DF
SS
Regression
1
3345.1
Error
7
9.8
Total
8
3354.9
a)
MS
3345.1
1.4
F
………
p
…………
The fastest pitchers can throw at about 100 mph. How far will such a pitch be hit? (2)
b) What is the value of R-squared? (2)
c) Fill in the F space in the ANOVA and explain specifically what is tested and what are
the conclusions. (3)
d) Is the constant (31.311) significant? Why? (2)
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3. (Render) A firm renovates homes in upstate New York. Sales and payroll for the region for a random
sample of years are given below. Sales are in $100,000 and Payroll is in $0.1 billions.
Current sales are $210000.00 and because of the opening of several new plants payroll is anticipated to
be about 0.6 billion dollars for the foreseeable future. Will average sales be significantly different from
current sales? To answer this question.
a) Complete the XY column (1)
b) Find the regression equation (4)
c) Find an appropriate interval for sales (3)
d) State and justify your conclusion (2)
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Y – Sales
2.0
3.0
2.5
2.0
2.0
3.5
15.0
X - Payroll
1
3
4
2
1
7
18
X2
1
9
16
4
1
49
80
Y2
4
9
6.25
4
4
12.25
39.50
XY
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4. The following data may look familiar. It is the service method data from the last exam. 11 service
methods were tried to see which is the fastest. I ran a multiple comparison of the 11 methods. Data were
stacked as ‘time’ in column 12 with the column labels ‘tel’ in column 13. This was run on Minitab in three
different ways, as a one-way ANOVA, using the Kruskal-Wallis test and using the Mood median test. I have
no idea how to do a Mood median test but I know this much. The null hypothesis is equal medians. The
assumptions are comparable to the Kruskal-Wallis test. The Mood median test is less powerful than the
Kruskal-Wallis test but is less affected by outliers. The Mood median test produces a chi-squared statistic
with degrees of freedom equal to the number of columns less 1.
Row
1
2
3
4
5
6
7
8
9
10
11
12
13
14
1
2.9
3.9
2.6
3.1
3.9
2.6
3.3
3.0
3.5
3.1
3.2
2.4
4.0
4.3
2
2.8
2.6
2.6
2.9
2.9
2.8
2.3
2.4
2.0
2.5
2.4
2.0
4.1
4.3
3
2.6
2.7
3.2
2.8
3.6
2.1
2.3
2.6
2.6
2.9
2.4
2.3
0.5
4.7
4
7.7
2.9
4.3
2.7
3.4
4.4
5.5
3.4
3.4
3.5
4.0
3.4
4.1
3.8
5
2.4
13.4
5.8
1.5
9.8
2.7
2.7
4.5
2.3
5.8
4.8
4.2
5.8
6.1
6
6.6
3.7
9.7
1.9
10.1
4.5
2.9
9.9
3.0
31.5
3.5
5.3
9.8
5.3
7
3.5
8.4
4.3
3.3
11.9
3.7
3.0
2.9
3.6
5.4
4.4
3.0
4.3
5.4
8
3.4
8.3
4.2
3.2
11.0
3.6
2.9
2.8
3.5
5.3
4.3
2.9
4.2
5.3
9
3.5
3.4
3.8
3.4
3.6
3.5
4.8
3.5
5.3
3.7
3.4
3.6
3.8
3.8
10
2.3
6.9
3.3
5.3
3.0
3.3
6.1
3.1
2.6
4.4
15.0
6.9
2.1
10.4
11
3.4
4.4
3.1
3.6
4.4
3.1
3.8
3.5
4.0
3.6
3.7
2.9
4.5
4.8
The edited Minitab output follows. Note that the p-values are missing and for you to think about.
Results for: 252x06031-01.MTW
MTB > Name c72 "RESI1"
#Residuals are stored for Normality test.
MTB > Oneway c12 c13;
SUBC>
Residuals 'RESI1';
SUBC>
GNormalplot;
SUBC>
NoDGraphs.
One-way ANOVA: time versus tel
Source
DF
SS
MS
F
P
tel
10
284.43 28.44
………
…………
Error
143 1228.86
8.59
Total
153 1513.28
S = 2.931
R-Sq = 18.80%
R-Sq(adj) = 13.12%
Level
tel 1
tel 10
tel 11
tel 2
tel 3
tel 4
tel 5
tel 6
tel 7
tel 8
tel 9
N
14
14
14
14
14
14
14
14
14
14
14
Mean
3.271
5.336
3.771
2.757
2.664
4.036
5.129
7.693
4.793
4.636
3.793
StDev
0.580
3.630
0.580
0.677
0.909
1.265
3.214
7.446
2.503
2.331
0.561
Individual 95% CIs For Mean Based on
Pooled StDev
------+---------+---------+---------+--(-----*-----)
(-----*------)
(-----*-----)
(-----*-----)
(------*-----)
(-----*-----)
(------*-----)
(-----*-----)
(-----*-----)
(------*-----)
(-----*-----)
------+---------+---------+---------+--2.5
5.0
7.5
10.0
Pooled StDev = 2.931
Normplot of Residuals for time (This plot was identical to the one shown below)
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MTB > Kruskal-Wallis c12 c13.
Kruskal-Wallis Test: time versus tel
Kruskal-Wallis Test on time
tel
N Median Ave Rank
tel 1
14
3.150
60.1
tel 10
14
3.850
86.8
tel 11
14
3.650
85.6
tel 2
14
2.600
33.3
tel 3
14
2.600
33.5
tel 4
14
3.650
85.4
tel 5
14
4.650
90.3
tel 6
14
5.300
107.2
tel 7
14
4.000
94.7
tel 8
14
3.900
89.5
tel 9
14
3.600
86.1
Overall 154
77.5
H = 41.97
Z
-1.53
0.82
0.72
-3.89
-3.87
0.70
1.12
2.61
1.51
1.06
0.75
#H is the Kruskal-Wallis statistic that I
#taught you about.
MTB > Mood c12 c13.
Mood Median Test: time versus tel
Mood median test for time
Chi-Square = 23.34
DF = 10
tel
tel
tel
tel
tel
tel
tel
tel
tel
tel
tel
tel
1
10
11
2
3
4
5
6
7
8
9
N<=
10
7
5
12
12
7
5
4
5
6
6
N>
4
7
9
2
2
7
9
10
9
8
8
Median
3.15
3.85
3.65
2.60
2.60
3.65
4.65
5.30
4.00
3.90
3.60
Q3-Q1
1.08
4.00
1.08
0.53
0.68
0.93
3.25
6.45
2.18
2.18
0.33
P = ………
Individual 95.0% CIs
-+---------+---------+---------+----(*--)
(--*------------)
(*--)
*-)
(*-)
(*-)
(-------*---)
(------*-----------------)
(--*-----)
(--*----)
*)
-+---------+---------+---------+----2.5
5.0
7.5
10.0
Overall median = 3.50
MTB > NormTest c72;
SUBC>
KSTest.
Probability Plot of RESI1
#This is a plot of the differences between means of
#each columns and the actual data.
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MTB > Vartest c12 c13;
SUBC>
Confidence 95.0.
Test for Equal Variances: time versus tel (This plot was not needed.)
Bartlett's Test (normal distribution)
Test statistic = 190.25, p-value = 0.000
Levene's Test (any continuous distribution)
Test statistic = 3.29, p-value = 0.001
a) Are the means significantly different? Identify the test that answers this question,
complete it and answer the question. Explain! (3)
b) Are the medians significantly different? Complete one or both of the remaining tests
and answer the question. Explain! (4)
c) At the end of the printout there are a K-S (actually Lilliefors) test, a Bartlett and a
Levene test. What do they tell us about the applicability of the ANOVA, Kruskal-Wallis
or Mood test to the problem? What conclusion is the most reliable? (3)
d) On the basis of all this, which of the service methods are best? (1)
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5. Four experts rated 4 brands of Columbian coffee. By adding together ratings on a seven point scale for
taste, aroma, richness and acidity, each coffee is given a rating on a 28 point scale. The following table
gives these summed ratings. Assume that the underlying distribution is not Normal and test for a difference
in the ratings of the four brands. Note:   .10 
Row
1
2
3
4
Brand A
Brand B
Brand C
Brand D
x1
x2
x3
x4
24
27
19
24
26
27
22
27
25
26
20
25
22
24
16
23
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ECO 252 QBA2
THIRD EXAM
Nov 30 and Dec 1, 2006
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. 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. Because so much of this exam is based on student numbers, there is a
penalty for failing to state your correct student number.
1) Bassett et al. give the following numbers for the year and the number of pensioners in the United
Kingdom. Pensioners are in millions. The 2000 number is a bit shaky, so subtract the last digit of your
student number divided by 10 from the 12.00 that you see there. Label your answer to this problem with a
version number. (Example: Good ol’ Seymour’s student number is 123456, so the 12.000 becomes 12.000 0.6 = 11.400 and he labels it Version 6.) 'Pensioners' is the dependent variable and 'Year' is the independent
variable, so what you are going to get is a trend line. If you don’t know what dependent and independent
variables are, stop work until you find out.
1
2
3
4
5
6
7
8
Year
1966
1971
1975
1976
1977
1978
1979
2000
Pensioners
6.679
7.677
8.321
8.510
8.637
8.785
8.937
12.000
Bassett et. al. strongly suggest that you change the base year to something other than the year zero. They
recommend that you subtract 1970 from every number in the ‘Year’ column, so that 1966 becomes -4 and
2000 becomes 30. This will make your computations easier.
a. Compute the regression equation Y  b0  b1 x to predict the number of pensioners in each year. (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 b1 and do a significance test on b1 (2)
e. Use your equation to predict the number of pensioners in 2005 and 2006. Using the 2006 number, create
a prediction interval for the number of pensioners for that year. Explain why a confidence interval for the
number of pensioners is inappropriate. (3)
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f. Make a graph of the data. Show the trend line clearly. If you are not willing to do this neatly and
accurately, don’t bother. (2)
g. What percent rise in pensioners did the equation predict for 2006? What percent rise does it predict for
2050? The population of the United Kingdom grew at roughly 0.31% a year over the last quarter of the 20 th
century. Can you intelligently guess what is wrong? (1)
[15]
2) The Lees in their text ask whether experience makes a difference in student earnings and present the
following data for student earnings versus years of work experience. To personalize these data, take the
second to last digit of your student number call it a . Clearly label the problem with a version number based
on your student number. Then take your a , multiply it by 0.5 and add it to the 13 in the lower left corner. .
(Example: Good ol’ Seymour’s student number is 123456, so the 13 becomes 13 + 0.5(5) = 13 + 2.5 = 15.5
and he labels it Version 5.) Each column is to be regarded as an independent random sample.
Years of Work Experience
1
2
3
16
19
24
21
20
21
18
21
22
13
20
25
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 experience matters or not. (4)
b) Using your results from a) present two different confidence intervals for the difference between earnings
for those with 1 and 3 years experience. Explain (i) under what circumstances you would use each interval
and (ii) whether the intervals show a significant difference. (2)
b) What other method could we use on these data to see if years of experience make a difference? Under
what circumstances would we use it? Try it and tell what it tests and what it shows. (3)
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c) (Extra Credit) Do a Levene test on these data and explain what it tests and shows. (4)
3) (Abronovic) A group of 4 workers produces defective pieces at the rates shown below during different
times of the day. Personalize the data by subtracting the last digit of your student number from the 14 in the
lower right corner. Use the number subtracted to label this as a version number. (Example: Good ol’
Seymour’s student number is 123456, so the 14 becomes 14 - 6 =8 and he labels it Version 6.)
Time
Worker’s Name
Apple
Plum
Pear
Melon
Early
10
11
8
12
Morning
Late
9
8
7
10
Morning
Early
12
13
11
11
Afternoon
Late
13
14
10
14
Afternoon
Sum of Row 1 = 41, SSQ of Row 1 = 429, Sum of Column 1 = 44, SSQ of Column 1 = 494,
Sum of Row 2 = 34, SSQ of Row 2 = 294, Sum of Column 2 = 46, SSQ of Column 2 = 550,
Sum of Row 3 = 47, SSQ of Row 3 = 555, Sum of Column 3 = 36, SSQ of Column 3 = 334,
a) Do a 2-way ANOVA on these data and explain what hypotheses you test and what the conclusions are.
(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)
c) What other method could we use on these data to see if time of day makes a difference while allowing for
cross-classification? Under what circumstances would we use it? Try it and tell what it tests and what it
shows. (3)
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