252x0771 11/26/07 (Page layout view!)
ECO252 QBA2
THIRD EXAM
November 29, 2007
Version 1
Name ______________________
Student number_______________
Class Day and hour____________
I. (8 points) Do all the following (2 points each unless noted otherwise). Make Diagrams! Show your
work!
x ~ N 26,14 
1. P20  x  38 
2. Px  0
3. P32  x  76 
4.
x.075
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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. Note that this is extremely long and that no one will do all the problems, so look
them over!
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. In an ordinary 1-way ANOVA, if the computed F statistic is below the value from the F table at the
given significance level, we can
a. Reject the null hypothesis because the difference between the means is not significant
b. Reject the null hypothesis because there is evidence of a significant difference between some of
the means.
c. Not reject the null hypothesis because the difference between the means is not significant.
d. Not reject the null hypothesis because the difference between the means is significant.
c. Not reject the null hypothesis because the difference between the variances is not significant.
d. Not reject the null hypothesis because the difference between the variances is significant.
e. None of the above.
[7]
3. After an analysis if variance, you would use the Tukey-Kramer procedure or similar confidence
intervals to check
a. For Normality
b. For equality of variances
c. For independence of error terms
d. For pairwise differences in means
e. For all of the above
f. For none of the above
4. If an ordinary one-way ANOVA has 25 columns 17 rows and 17 25   425 , the degrees of freedom
for the F test are
a. 400 and 24
b. 408 and 16
c. 24 and 400
d. 16 and 408
e. 400 and 424
f. 408 and 424
g. 424 and 400
h. 424 and 408
i. 16 and 24
j. None of the above. The correct answer is _______.
5. Assuming that your answer to 4 is correct and that the significance level is 5%, the correct value of F
from the table is _______. (This may have to be approximate. If so, what did you use?) (1)
[12]
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Exhibit 1 A realtor believes that the selling price of a home (in $ thousands) is related to the condition of
the home (on a 1 to 10 scale) and the size of the home (in hundreds of square feet). He runs the data below
on Minitab and gets the following.
Row Price Size Condition
1
360
23
5
2
200
11
2
3
340
20
9
4
280
17
3
5
280
15
8
6
330
21
4
7
380
24
7
8
250
13
6
MTB > regress c1 2 c2 c3
Regression Analysis: Price versus Size, Condition
The regression equation is
Price = 64.5 + 11.7 Size + 4.88 Condition
Predictor
Coef SE Coef
T
P
Constant
64.539
4.228 15.27 0.000
Size
11.7282
0.2317 50.62 0.000
Condition
4.8826
0.4494 _____ _____
S = 2.75997
R-Sq = 99.9%
R-Sq(adj) = 99.8%
Analysis of Variance
Source
DF
SS
Regression
2 25712
Residual Error
5
38
Total
7 25750
Source
Size
Cond
DF
1
1
MS
12856
8
F
1687.70
P
0.000
Seq SS
24813
899
The sum of the price column is 2420 and the sum of the squared numbers in the sales column is not needed.
The sum of the 'Size' column is 144 and the sum of the squared numbers in the Size column is 2750.
The sum of the ‘Condition’ column is 44 and the sum of the squared numbers in the Condition column is
284.
If Price is the dependent variable and Size and Condition are the independent variables we have found that
the sum of x1y is 45540 and the sum of x1 x2 is 818. The sum of x2y has not been computed.
6 and 7. In the multiple regression, are the coefficients of size and condition significant at the 5%
significance level? Give reasons. Do not do unneeded computations. (2)
[15]
8. Assuming that the coefficients in the multiple regression are correct, what price would we predict for a
home with 20(hundred) square feet and a condition score of 9? (1)
9. Using the information in the multiple regression printout, make your result in 8) into a rough prediction
interval. (2)
10. Using the information in the printout, what is the value of R-squared for a regression of ‘Price’ against
‘Size’ alone? (2)
[20]
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Exhibit 1 A realtor believes that the selling price of a home
(in $ thousands) is related to the condition of the home (on a
1 to 10 scale) and the size of the home (in hundreds of square
feet). He runs the data below on Minitab and gets the
following.
Row Price Size Condition
1
360
23
5
2
200
11
2
3
340
20
9
4
280
17
3
5
280
15
8
6
330
21
4
7
380
24
7
8
250
13
6
MTB > regress c1 2 c2 c3
Regression Analysis: Price versus Size, Condition
The regression equation is
Price = 64.5 + 11.7 Size + 4.88 Condition
Predictor
Coef SE Coef
T
P
Constant
64.539
4.228 15.27 0.000
Size
11.7282
0.2317 50.62 0.000
Condition
4.8826
0.4494 _____ _____
S = 2.75997 R-Sq = 99.9% R-Sq(adj) =99.8%
Analysis of Variance
Source
DF SS
MS
F
P
Regression
2 25712 12856 1687.70 0.000
Residual Erro 5
38
8
Total
7 25750
Source
DF Seq SS
Size
1
24813
Condition 1
899
The sum of the price column is 2420 and the sum of the squared numbers in the sales column is not needed.
The sum of the 'Size' column is 2750 and the sum of the squared numbers in the Size column is 2950.
The sum of the ‘Condition’ column is 44 and the sum of the squared numbers in the Condition column is 284.
If Price is the dependent variable and Size and Condition are the independent variables we have found that the sum of x1y is 45540
and the sum of x1 x2 is 818. The sum of x2y has not been computed.
11. Do a simple regression of ‘Price’ against ‘Condition’ alone.
xy that you will need for this regression. Show your work! (2)
a) Compute the sum

Don’t compute stuff that has already been done for you!
b) It says that you do not need to know the sum of squares in the sales column. You do
Y 2  nY 2 . Without doing any computing, tell
however need the spare part SS y 

what its value is. (1)
c) Compute the coefficients of the equation Yˆ  b0  b2 x to predict the value of ‘Price’ on
the basis of ‘Condition.’ (4)
[27]
d) Compute R 2 . (3)
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Exhibit 1 A realtor believes that the selling price of a home
(in $ thousands) is related to the condition of the home (on a
1 to 10 scale) and the size of the home (in hundreds of square
feet). He runs the data below on Minitab and gets the
following.
Row Price Size Condition
1
360
23
5
2
200
11
2
3
340
20
9
4
280
17
3
5
280
15
8
6
330
21
4
7
380
24
7
8
250
13
6
MTB > regress c1 2 c2 c3
Regression Analysis: Price versus Size, Condition
The regression equation is
Price = 64.5 + 11.7 Size + 4.88 Condition
Predictor
Coef SE Coef
T
P
Constant
64.539
4.228 15.27 0.000
Size
11.7282
0.2317 50.62 0.000
Condition
4.8826
0.4494 _____ _____
S = 2.75997 R-Sq = 99.9% R-Sq(adj) =99.8%
Analysis of Variance
Source
DF SS
MS
F
P
Regression
2 25712 12856 1687.70 0.000
Residual Erro 5
38
8
Total
7 25750
Source
DF Seq SS
Size
1
24813
Condition 1
899
The sum of the price column is 2420 and the sum of the squared numbers in the sales column is not needed.
The sum of the 'Size' column is 144 and the sum of the squared numbers in the Size column is 2750.
The sum of the ‘Condition’ column is 44 and the sum of the squared numbers in the Condition column is 284.
If Sales is the dependent variable and Size and Condition are the independent variables we have found that the sum of x1y is 45540
and the sum of x1 x2 is 818. The sum of x2y has not been computed.
e) Is the slope of the simple regression significant at the 5% level? Do not answer this question
without appropriate calculations! (4)
f) Predict the price of an average home with a condition of 9 and make your estimate into an
appropriate 99% interval. (4)
g) Do an analysis of variance using your SST, SSE and SSR for this equation or using 1,
R 2 and 1  R 2 . What have you already done that makes this table redundant? If you
don’t know what redundant means, ask! (3)
[43]
h) Using the information on Regression Sums of squares or R 2 and 1  R 2 in the
ANOVA that you just did and from the multiple regression, do an F test to see if adding
‘Size’ to the regression of ‘Price’ against ‘Condition’ is worthwhile. Do not waste our time by
repeating stuff that has already been done. (3)
[46]
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Exhibit 2 (Groebner) A product is being produced on 3 different lines using 3 different layouts for the
lines. A sample of 36 observations are taken on various days over a period of four weeks so that there are
12 observations for the daily output for each line evenly divided between the three possible layouts. Assume
  .05 .
MTB > Twoway c4 c2 c3;
SUBC>
Means c2 c3.
Two-way ANOVA: output 1 versus line, layout
Source
DF
line
2
layout
2
Interaction __
Error
__
Total
35
S = 20.63
R-Sq
SS
MS
F
P
187.1
93.5
0.22 0.804
28263.4 14131.7 33.21 0.000
_______
_____
____ _____
11489.0
425.5
41874.6
= 72.56%
R-Sq(adj) = 64.43%
Individual 95% CIs For Mean Based on
Pooled StDev
line
Mean ------+---------+---------+---------+--1
132.583
(---------------*--------------)
2
128.167
(--------------*--------------)
3
127.417 (--------------*---------------)
------+---------+---------+---------+--120.0
128.0
136.0
144.0
Individual 95% CIs For Mean Based on
Pooled StDev
layout
Mean ----+---------+---------+---------+----1
116.667
(----*----)
2
168.250
(----*----)
3
103.250 (----*----)
----+---------+---------+---------+----100
125
150
175
12. Fill in the missing degrees of freedom, the missing sum of squares and the missing mean square. (2)
[48]
13. Is there significant interaction between ‘line’ and ‘layout’? Don’t answer unless you can tell me what the
evidence is. (2)
14. Is the difference between lines significant? Why?(1)
15. Do a confidence interval of your choice for the difference between layout 1 and layout 3. Tell what kind
of interval you are using , what its characteristics are and whether it shows a significant difference. (4)[55]
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16. (Groebner) An industrial firm analyses the amount of breakage (in dollar cost) that occurs using 3
different shipping methods. There is a strong likelihood that the data does not come from the Normal
distribution. The purpose of the test is to see if the four shipping methods differ in breakage. The columns
can be considered random samples.
Rail
Plane Truck
7960
8053
8818
8399
7764
9432
9429
9196
9260
6022
5821
5676
The most appropriate method for doing this test is:
a) The Friedman Test
b) The Kruskal-Wallis Test
c) One-way ANOVA
d) Two-way ANOVA
e) The sign test
[57]
f) Another test (Name it!)
17. Assume that your decision is correct in 16. What is your null hypothesis or hypotheses? Be specific! Are
you talking about rows or columns or both? Are you comparing means, medians, proportions or variances?
18. OK. Let’s see you do the test. (4)
[63]
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ECO252 QBA2
THIRD EXAM
Nov 26-29, 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 1% 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. You should explain whether the coefficients are significant at the 1% level. 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. (3)
III Do the following. (22+ 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. Show your work!
1) The Lees, in their book on statistics for Finance majors, ask about the relationship of gasoline prices  y 
in cents per gallon to crude oil prices x1  in dollars per barrel and present the data for the years 1975 1988. I have obtained most of the data for the years 1980 – 2007. It is presented below.
Row
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
Year
1980
1981
1982
1983
1984
1985
1986
1987
1988
1989
1990
1991
1992
1993
1994
1995
1996
1997
1998
1999
2000
2001
2002
2003
2004
2005
2006
2007
GasPrice
1.25
1.38
1.30
1.24
1.21
1.20
0.93
0.95
0.96
1.02
1.16
1.14
1.13
1.11
1.11
1.15
1.23
1.23
1.06
1.17
1.51
1.46
1.36
1.59
1.88
2.30
*
3.10
CrudePrice
26.07
35.24
31.87
26.99
28.63
26.25
14.55
17.90
14.67
17.97
22.22
19.06
18.43
16.41
15.59
17.23
20.71
19.04
12.52
17.51
28.26
22.95
24.10
28.53
36.98
50.23
*
90.00
Yr-1979
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
This data set also contains the year with 1979 subtracted from it x 2  . You may need to use this later.
Ignore it in Problem 1. Note that the numbers for 2006 have not yet been published in my source, Statistical
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Abstract of the United States, and that the numbers for 2007 are my estimates for third quarter prices. These
are unleaded prices, which the Lees did not use. You are supposed to use only the numbers for 1990
through 2006 and one other observation for your data. You will thus have n  17 observations. The other
column is the value for the year 1980  a  , where a is the second to last digit of your student number. If
you are unsure of the data that you are using or if you want help with the sums that you need to do the
regression go to 3takehome072a.
Show your work – it is legitimate to check your results by running the problem on the computer. (In fact, I
will give you 2 points extra credit for checking it and annotating the output for significance tests etc.) But I
expect to see hand computations for every part of this problem.
a. Compute the regression equation Y  b0  b1 x to predict the price of gasoline on the basis of
crude oil prices. (3)
b. Compute R 2 . (2)
c. Compute s e . (2)
d. Compute s b1 and do a significance test on b1 (2)
e. Compute a confidence interval for b0 . (2)
f. You have a crude price for 2007. Using this, predict the gasoline price for 2007 and create a
prediction interval for the price of gasoline for that year. Explain why a confidence interval for the
price is inappropriate and check to see if my estimated price is in the interval. (3)
g. Do an ANOVA for this regression. (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)
[19]
2) Now we can use the date to see if there is a trend line in addition to the effect of crude oil.
a. Do a multiple regression of the price of gasoline against crude prices and the data variable,
which has been massaged to make 1980 year 1. This involves a simultaneous equation solution.
Attempting to recycle b1 from the previous page won’t work. (7)
c. Compute the regression sum of squares and use it in an ANOVA F test to test the usefulness of
this regression. (4)
b. Compute R 2 and R 2 adjusted for degrees of freedom for both this and the previous problem.
Compare the values of R 2 adjusted between this and the previous problem. Use an F test to
compare R 2 here with the R 2 from the previous problem. The F test here is one to see if adding a
new independent variable improves the regression. This can also be done by modifying the
ANOVAs in b.(4)
d. Use your regression to predict the price of gasoline in 2007. Is this closer to the estimated
gasoline price? Do a confidence interval and a prediction interval. (3)
[37]
e. Again there is extra credit for checking your results on the computer. Use the pull-down menu or
try
Regress GasPrice on 2 CrudePrice Yr-1979 (2)
3) According to Russell Langley, three sopranos were discussing their recent performances. Fifi noted that
she got 36 curtain calls at La Scala last week, but Adalina put her down with the fact that she got 39. Could
one of the singers really say that she had more curtain calls than another or could the differences just be due
to chance?
Personalize the data below by adding the last digit of your student number to each number in the
first row. Use a 10% significance level throughout this question.
Row
1
2
3
4
Fifi
36
22
19
16
Adelina
39
14
20
18
Maria
21
32
28
22
a) State your hypothesis and use a method to compare means assuming that each column represents a
random sample of curtain calls at La Scala. (4)
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b) Still assuming that these are random samples, use a method that compares medians instead. (3)
c) Actually, these were not random samples. Though row 1 represents curtain calls at La Scala (Milan), row
2 was in Venice, row 3 in Naples and row 4 in Rome. Will this affect our results? Does this show anything
about audiences on the four cities? Use an appropriate method to compare medians. (5)
d) Do two different types of confidence intervals between Milan and the least enthusiastic opera house.
Explain the difference between the intervals. (2)
e) Assume that we want to compare medians instead. How does the fact that these data were collected at
three opera houses affect the results? (3)
f) Do you prefer the methods that compare medians or means? Don’t answer this unless you can
demonstrate an informed opinion. (1)
g) (Extra credit) Do a Levine test on these data and explain what it tests and shows.(3)
h) (Extra credit)Check your work on the computer. This is pretty easy to do. Use the same format as in
Computer Problem 2, but instead of car and driver numbers use the singers’ and cities’ names. You can use
the stat and ANOVA pull-down menus for One-way ANOVA, two-way ANOVA and comparison of
variances of the columns. You can use the stat and the non-parametrics pull-down menu for Friedman and
Kruskal-Wallis. You also probably ought to test columns for Normality. Use the Statistics pull-down menu
and basic statistics 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 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) [58]
You should note conclusions on the printout – tell what was tested and what your conclusions are using a
10% significance level.
11