252y0771t 11/28/07
Warning – This document is 74 pages. At most you need the first 16
pages and about 5 pages from the two appendices for your individual
solutions.
1
252y0771t 11/28/07
ECO252 QBA2
THIRD EXAM
Nov 26-29, 2007
TAKE HOME SECTION
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!
 y  in cents per gallon to
x1  in dollars per barrel and present the data for the years 1975 - 1988. I have obtained most of the data for the
1) The Lees, in their book on statistics for Finance majors, ask about the relationship of gasoline prices
crude oil prices
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
This data set also contains the year with 1979 subtracted from it
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
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 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
row 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)
2
b. Compute R . (2)
c. Compute s e . (2)
2
252y0771t 11/28/07
d. Compute
s b1
and do a significance test on
e. Compute a confidence interval for
b1 (2)
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)
b. Compute the regression sum of squares and use it in an ANOVA F test to test the usefulness of
c. Compute
R2
and
R2
this regression. (4)
adjusted for degrees of freedom for both this and the previous problem. Compare the values
2
2
2
of R adjusted between this and the previous problem. Use an F test to
compare R here with the R 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)
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.
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. The original data set is presented above.
3
252y0771t 11/28/07
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
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
row 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. The 10 data sets follow.
Version 0
Row
GP0
1 1.25
2 1.16
3 1.14
4 1.13
5 1.11
6 1.11
7 1.15
8 1.23
9 1.23
10 1.06
11 1.17
12 1.51
13 1.46
14 1.36
15 1.59
16 1.88
17 2.30
CP0
26.07
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
Yr0
1
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
Version 1
Row
GP1
1 1.38
2 1.16
3 1.14
4 1.13
5 1.11
6 1.11
7 1.15
8 1.23
9 1.23
10 1.06
11 1.17
12 1.51
13 1.46
14 1.36
15 1.59
16 1.88
17 2.30
CP1
35.24
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
Yr1
2
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
Version 2
Row
GP2
1 1.30
2 1.16
3 1.14
4 1.13
5 1.11
6 1.11
7 1.15
8 1.23
9 1.23
10 1.06
11 1.17
12 1.51
13 1.46
14 1.36
15 1.59
16 1.88
17 2.30
CP2
31.87
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
Yr2
3
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
Version 3
Row
GP3
1 1.24
2 1.16
3 1.14
4 1.13
5 1.11
6 1.11
7 1.15
8 1.23
9 1.23
10 1.06
11 1.17
12 1.51
13 1.46
14 1.36
15 1.59
16 1.88
17 2.30
CP3
26.99
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
Yr3
4
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
Version 4
Row
GP4
1 1.21
2 1.16
3 1.14
4 1.13
5 1.11
6 1.11
7 1.15
8 1.23
9 1.23
10 1.06
11 1.17
12 1.51
13 1.46
14 1.36
15 1.59
16 1.88
17 2.30
CP4
28.63
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
Yr4
5
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
Version 5
Row
GP5
1 1.20
2 1.16
3 1.14
4 1.13
5 1.11
6 1.11
7 1.15
8 1.23
9 1.23
10 1.06
11 1.17
12 1.51
13 1.46
14 1.36
15 1.59
16 1.88
17 2.30
CP5
26.25
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
Yr5
6
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
4
252y0771t 11/28/07
Version 6
Row
GP6
1 0.93
2 1.16
3 1.14
4 1.13
5 1.11
6 1.11
7 1.15
8 1.23
9 1.23
10 1.06
11 1.17
12 1.51
13 1.46
14 1.36
15 1.59
16 1.88
17 2.30
CP6
14.55
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
Yr6
7
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
Version 9
Row
GP9
1 1.02
2 1.16
3 1.14
4 1.13
5 1.11
6 1.11
7 1.15
8 1.23
CP9
17.97
22.22
19.06
18.43
16.41
15.59
17.23
20.71
Yr9
10
11
12
13
14
15
16
17
Version 7
Row
GP7
1 0.95
2 1.16
3 1.14
4 1.13
5 1.11
6 1.11
7 1.15
8 1.23
9 1.23
10 1.06
11 1.17
12 1.51
13 1.46
14 1.36
15 1.59
16 1.88
17 2.30
9
10
11
12
13
14
15
16
17
1.23
1.06
1.17
1.51
1.46
1.36
1.59
1.88
2.30
Minitab computed the following Descriptive Statistics.
Variable
N N*
Mean SE Mean
StDev
Year
28
0 1993.5
1.55
8.23
GasPrice
27
1 1.3381
0.0880 0.4571
CrudePrice 27
1
25.92
2.94
15.28
Yr-1979
28
0
14.50
1.55
8.23
GP0
18
1
1.441
0.123
0.522
CP0
18
1
26.99
4.25
18.05
Yr0
19
0
18.53
1.54
6.70
GP1
18
1
1.448
0.123
0.520
CP1
18
1
27.50
4.28
18.15
Yr1
19
0
18.58
1.50
6.56
GP2
18
1
1.444
0.123
0.521
CP2
18
1
27.31
4.26
18.08
Yr2
19
0
18.63
1.47
6.42
GP3
18
1
1.441
0.123
0.522
CP3
18
1
27.04
4.25
18.05
Yr3
19
0
18.68
1.44
6.29
GP4
18
1
1.439
0.123
0.523
CP4
18
1
27.13
4.25
18.05
Yr4
19
0
18.74
1.41
6.16
GP5
18
1
1.438
0.123
0.523
CP5
18
1
27.00
4.25
18.05
Yr5
19
0
18.79
1.39
6.04
GP6
18
1
1.423
0.126
0.534
CP6
18
1
26.35
4.31
18.29
Yr6
19
0
18.84
1.36
5.93
GP7
18
1
1.424
0.126
0.533
CP7
18
1
26.54
4.28
18.18
Yr7
19
0
18.89
1.34
5.82
GP8
18
1
1.425
0.126
0.533
CP8
18
1
26.36
4.31
18.28
Yr8
19
0
18.95
1.31
5.72
GP9
18
1
1.428
0.125
0.530
CP9
18
1
26.54
4.28
18.17
Yr9
19
0
19.00
1.29
5.63
CP7
17.90
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
Yr7
8
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
19.04
12.52
17.51
28.26
22.95
24.10
28.53
36.98
50.23
18
19
20
21
22
23
24
25
26
Minimum
1980.0
0.9300
12.52
1.00
1.060
12.52
1.00
1.060
12.52
2.00
1.060
12.52
3.00
1.060
12.52
4.00
1.060
12.52
5.00
1.060
12.52
6.00
0.930
12.52
7.00
0.950
12.52
8.00
0.960
12.52
9.00
1.020
12.52
10.00
Version 8
Row
GP8
1 0.96
2 1.16
3 1.14
4 1.13
5 1.11
6 1.11
7 1.15
8 1.23
9 1.23
10 1.06
11 1.17
12 1.51
13 1.46
14 1.36
15 1.59
16 1.88
17 2.30
Q1
1986.3
1.1100
17.51
7.25
1.138
17.44
14.00
1.138
17.44
14.00
1.138
17.44
14.00
1.138
17.44
14.00
1.138
17.44
14.00
1.138
17.44
14.00
1.125
17.03
14.00
1.125
17.44
14.00
1.125
17.03
14.00
1.125
17.44
14.00
Median
1993.5
1.2100
22.22
14.50
1.230
21.47
19.00
1.230
21.47
19.00
1.230
21.47
19.00
1.230
21.47
19.00
1.220
21.47
19.00
1.215
21.47
19.00
1.200
19.89
19.00
1.200
19.89
19.00
1.200
19.89
19.00
1.200
19.89
19.00
CP8
14.67
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
Yr8
9
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
Q3
2000.8
1.3800
28.53
21.75
1.530
28.33
24.00
1.530
30.21
24.00
1.530
29.37
24.00
1.530
28.33
24.00
1.530
28.56
24.00
1.530
28.33
24.00
1.530
28.33
24.00
1.530
28.33
24.00
1.530
28.33
24.00
1.530
28.33
24.00
In the following, Calculations will be done for version 0 only. Spare parts and Solutions for all other
versions will follow in Appendix A. Data follows with the labels given to the columns in Minitab shown.
The first set of sums is from Minitab. The second set of sums comes from my own little 30-year old
calculator, but they were checked against Minitab.
5
252y0771t 11/28/07
X1
Y
Row
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
GP0_1 CP0_1
1.25 26.07
1.16 22.22
1.14 19.06
1.13 18.43
1.11 16.41
1.11 15.59
1.15 17.23
1.23 20.71
1.23 19.04
1.06 12.52
1.17 17.51
1.51 28.26
1.46 22.95
1.36 24.10
1.59 28.53
1.88 36.98
2.30 50.23
22.84 395.84
X 12
Y2
X2
X 22
Yr0_1
Ysq
X1sq
1 1.5625
679.64
11 1.3456
493.73
12 1.2996
363.28
13 1.2769
339.66
14 1.2321
269.29
15 1.2321
243.05
16 1.3225
296.87
17 1.5129
428.90
18 1.5129
362.52
19 1.1236
156.75
20 1.3689
306.60
21 2.2801
798.63
22 2.1316
526.70
23 1.8496
580.81
24 2.5281
813.96
25 3.5344 1367.52
26 5.2900 2523.05
297 32.4034 10551.0
X2sq
1
121
144
169
196
225
256
289
324
361
400
441
484
529
576
625
676
5817
X1 Y
X 2Y
X1 X 2
X1Y
X2Y
32.588
1.25
25.775 12.76
21.728 13.68
20.826 14.69
18.215 15.54
17.305 16.65
19.815 18.40
25.473 20.91
23.419 22.14
13.271 20.14
20.487 23.40
42.673 31.71
33.507 32.12
32.776 31.28
45.363 38.16
69.522 47.00
115.529 59.80
578.272 419.63
X1X2
26.07
244.42
228.72
239.59
229.74
233.85
275.68
352.07
342.72
237.88
350.20
593.46
504.90
554.30
684.72
924.50
1305.98
7328.80
You were given the sums of these columns with the first number missing. Unfortunately, one of these was
wrong. The fact that I received no complaints before Wednesday indicates either 1) you didn’t use them or
Y  21 .59 ,
2) you didn’t know that R 2 is between zero and 1. The sums appear below.

 Y  30.8409 ,  X 1  369.77 ,   9871 .3365 ,  X 2  296 ,  X
 X Y  545.6841,  X Y  418.38 and  X X  7302.73 .
2
X 12
1
2
1
2
2
 5816 ,
2
You could have computed the following.
 Y  21.59  1.25  22.84 ,  Y  30.8409  1.25  32.4034 ,  X 1  369.77  26.07  395.84 ,
 X  9871 .3365  26.07  10550 .9814 ,  X 2  296  1  297 ,  X  5816  1  5817 ,
 X 1 Y  545.6841 26.071.25  578.2716 ,  X 2 Y  418.38 11.25  419.63 and
 X 1 X 2  7302.73  26.071  7328.80 . Remember that n  17 . I didn’t.
2
2
1
2
2
2
2
2
I will use the second set of sums in my calculations.
Spare Parts Computation:
22 .84
395 .84
297
Y 
 1.34353 , X 1 
 23 .28471 , X 2 
 17 .47059 †
17
17
17
X
X
Y
2
1
 nX 12  SSX1  10550.9814  1723.284712  1333.9602 *
2
2
 nX 22  SSX 2  5817  17 17 .47059 2  628 .2343 *†
2
 nY 2  SST  SSY  32 .4034  17 1.34353 2  1.71716 *
 X Y  nX Y  SX Y  578 .2716 1723.28471 1.34353   46.4486
 X Y  nX Y  SX Y  419 .63 1717.47059 1.34353   20.6016 †
 X X  nX X  SX X  7328 .80 1723.28471 17.47059   413 .242 †
1
1
2
2
1
2
1
2
1
2
1
2
†Needed only in next problem. *Must be positive. The rest may well be negative.
6
252y0771t 11/28/07
a. Compute the regression equation Y  b0  b1 x to predict the price of gasoline on the basis of
crude oil prices. (3)
22 .84
395 .84
 1.34353 , X 1 
 23 .28471
Solution: First copy n  17 Y 
17
17
X
Y
2
1
2
 nX 12  SSX1  10550.9814  1723.284712  1333.9602 *
 nY 2  SST  SSY  32 .4034  17 1.34353 2  1.71716 *
 X Y  nX Y  SX Y  578 .2716 1723.28471 1.34353   46.4486
S xy
 XY  nXY  46.4486  0.03482 and b  Y  b X  1.34353  0.03482  23.28471 
So b1 

SS x  X 2  nX 2 1333 .9602
1
1
1
0
1
 0.53276 , which means Yˆ  0.53276  0.03482 X or Y  0.53276  0.03482 X  e
b. Compute R 2 . (2)
Solution: SSR  b1 S xy  0.0348246.4486  1.61734. We can say R 2 
R2 
b1 S xy
SSy

SSR 1.61734

 .94167 or
SST 1.71716
S xy 2
0.03482 46 .4486 
46 .4486 2

 .94187
 .94167 or R 2 
1333 .9602 1.71716 
1.71716
SS x SS y
c. Compute s e . (2)
Solution: We can compute SSE  SST  SSR  1.71716  1.61734  0.09982 . Then
SS y  b1 S xy 1.71716  0.03482 46 .4484 
SSE 0.09982
s e2 

 0.006655 or s e2 

 0.006655
n2
15
n2
15
s
e

 0.006655  0.081576 .
d. Compute s b1 and do a significance test on b1 (2)
 1  0.006655
So s b21  s e2 
 0.0000050 and s b  .0000050  0.0022336 . The outline says to test

1
 SS x  1333 .9612
b 0
H 0 : 1  0
use t  1
and if the null hypothesis is false in that case we say that 1 is significant. So

H
:


0
s b1
 1 1
0.03462  0
0.0022338
 15 .480 is outside both these zones, so that there is no doubt that the coefficient is significant.
15
 2.131 if   .05 . Our calculated t 
our ‘do not reject’ zone is between  t .025
e. Compute a confidence interval for b0 . (2).
 1 23 .28471 2 
1 X 2 



0
.
006655
Solution: Recall X 1  23 .28471 s b20  s e2  
 


 n SS x 
17 1333 .9612 
 0.006655 0.465265   0.003096 . So sb0  0.003096  0.05564
A confidence interval would be  0  b0  t  2 sb0  0.53276 2.1310.05564  0.53276 0.11858
Notice that since this interval does not include zero, the constant is significant.
7
252y0771t 11/28/07
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)
1 X X 2

Solution: The prediction Interval is Y0  Yˆ0  t sY , where sY2  s e2   0
 1 . The crude price
n

SS x


ˆ
was 90, so Y  0.53276  0.03482 X  0.53276  0.03482 90   3.67 and


 1 90  23 .28471 2

sY2  0.006655  
 1  0.006655 0.0588235  3.3366262  1
 17

1333 .9612


 0.006655 4.3954497   0.029252 sY  0.0292517  0.1710 . So the prediction interval is
Y0  3.67  2.1310.1710   3.67  0.36 or 3.31 to 4.03. The interval seems too high, but wait a few weeks.
g. Do an ANOVA for this regression. (3)
Y 2  nY 2  SST  SSY  1.71716 ,
Solution: It’s time to repeat our recent results.

SSR  b1 S xy  1.61734 and SSE  SST  SSR  0.09982
So our ANOVA table will be as below.
Source
SS
DF
Regression
1.6173
1
MS
F
1.6173
243.09
F
1,15  4.54
F.05
Error
0.0998
15
0.006653
Total
1.7171
16
Since our computed F is larger than the table F, we reject the null hypothesis that the independent variable
and the dependent variable have no linear relation.
f) Make a graph of the data. Show the regression line and the data points clearly. If you are not
willing to do this neatly and accurately, don’t bother. (2)
[19]
8
252y0771t 11/28/07
2) Now we can use the data 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)
Solution: The spare parts computation is repeated from the previous problem.
Spare Parts Computation:
22 .84
395 .84
297
Y 
 1.34353 , X 1 
 23 .28471 , X 2 
 17 .47059
17
17
17
X
X
Y
2
1
 nX 12  SSX1  10550.9814  1723.284712  1333.9602 *
2
2
 nX 22  SSX 2  5817  17 17 .47059 2  628 .2343 *
2
 nY 2  SST  SSY  32 .4034  17 1.34353 2  1.71716 *
 X Y  nX Y  SX Y  578 .2716 1723.28471 1.34353   46.4486
 X Y  nX Y  SX Y  419 .63 1717.47059 1.34353   20.6016
 X X  nX X  SX X  7328 .80 1723.28471 17.47059   413 .242
1
1
2
2
1
2
1
2
1
2
1
2
*Must be positive. The rest may well be negative.
Simplified Normal Equations:
X 1Y  nX 1Y  b1
 X  nX   b  X X  nX X 

 X Y  nX Y  b  X X  nX X   b  X  nX  ,
2
Which are
2
1
2
1
1
2
1
2
2
1
2
1
2
2
1
2
2
2
2
2
46 .4486  1333 .9602 b1  413 .242 b2

20 .6016  413 .242 b1  628 .2343 b2
and solve them as two equations in two unknowns for b1 and b2 . These are a fairly tough pair of equations
to solve. To use elimination, you must make the coefficients of b1 or b2 equal and then subtract one
equation from the other. To do this either multiply the first equation by
628 .2343
 1.520257621 or
413 .242
413 .242
 0.309785854 . (You could multiply the second equation instead.
1333 .9602
But this is enough.) If we multiply the first equation by 1.520257621 we get
70 .6138  2027 .963 b1  628 .2343 b2
. If we now subtract the second equation from the first, we get

20 .6016  413 .242 b1  628 .2343 b2
50 .0122  1614 .721b1 or, if we divide through by 1414.721, b1  0.03097 . We can now substitute this
multiply the first equation by
value in one of our original equations and solve for b2 . The second of these equations can be rearranged to
give us 628 .2343 b2  20.6016  413 .242 b1 .
If we divide through by 628.2343, we get b2  0.0327929  0.6577832 b1 or
b2  0.0327929  0.6577832 0.03097  or b2  0.01242 . Finally we have b0  Y  b1 X 1  b2 X 2 ,
Y  1.34353, X 1  23 .28471 and X 2  17 .47059 . So
b0  1.3435  0.03097 23.28471   0.01242 17.47059   1.3435  0.7211  0.2170  0.4054
14 .389  413 .242 b1  128 .017 b2
Note: If I multiply by 0.309785854, using Minitab as a calculator, I get 
.
20 .603  413 .242 b1  628 .234 b2
When I subtracted the top equation from the bottom, I got 6.212  500 .218 b2 . This led to b2  0.01242 .
Unfortunately, since Minitab does not display all the digits with which it is working, you may not get quite
the same answers.
9
252y0771t 11/28/07
To summarize, b0  0.4054 , b1  0.03097 and b2  0.01242 , so our equation is
Yˆ  b0  b1 X 1  b2 X 2  0.4054  0.03097X 1  0.01242X 2
b. Compute the regression sum of squares and use it in an ANOVA F test to test the usefulness of
this regression. (4)
Y 2  nY 2  SST  SSY  1.71716 ,
Solution: We have already found b1  0.03097 , b2  0.01242 ,
 X Y  nX Y  SX Y  46.4486 and  X
1
1
1

2Y
 nX 2 Y  SX 2 Y  20 .6016 . The most appropriate
formula for the regression sum of squares for this second regression is below.
SSR2  b1 Sx1 y  b2 Sx2 y  0.03097 46.4486   0.01242 20.6016   1.43851  0.25587  1.6944
SSE  SST  SSR  1.7172  1.6944  0.0228
Source
Regression
SS
1.6944
DF
2
MS
0.8472
Fcalc
520.20s
F
2,14  3.74
F.05
Error
0.0228
14
0.0016286
Total
1.7172 16
The null hypothesis is no connection between Y and the X’s. It is rejected because the calculated F is above
the table value. Rounding error has affected our value of Fcalc . Fortunately, this rarely makes a difference.
c. 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)
Solution: For the first regression with one independent variable we had
SSR 1.61734

 .94167 . Our
SSR1  b1 S xy  0.0348246.4486  1.61734 and we had R 2  RY2.1 
SST 1.71716
ANOVA read
Source
SS
DF
MS
F
F
Regression
1.6173
1
1.6173
243.09
F 1,15  4.54
.05
Error
0.0998
15
0.006653
Total
1.7171
16
For the latest regression
SSR 1.6944
R 2  RY2.12 

 0.9867 . (This must be between zero and one!). If we use R 2 , which is
SST 1.7172
R 2 adjusted for degrees of freedom, for
the first regression, the number of independent variables was k  1 and n  k  1  17  1  1  15 .
n  1R 2  k  16  .94167  1  .9378 and for the second regression k  2 and
RY2.1 
n  k 1
15
n  1R 2  k  16 .9867  2  .9848 R-squared adjusted is supposed to
n  k  1  17  2  1  14 . RY1.12 
n  k 1
14
rise if our new variable has any explanatory power. It did.
10
252y0771t 11/28/07
Now it’s time to do the ANOVA. There are two ways to do it. We can take an ANOVA of the type used in
the last regression and take the regression sum of squares SSR2 and divide it into the regression sum of
squares from the first regression SSR1 and the additional sum of squares SSR 2  SSR1 . In the schemes
below k is the number of independent variables in the first regression and k  r is the number of
independent variables in the second regression.
Source
SS
DF
MS
Fcalc
F
k
First Regression SSR1
MSR1
SSR  SSR r
2nd Regression
MSR MSE F r , nk r 1
MSR
2
1
2
2
Error
n  k  r  1 MSE
SSE
Total
SST
n 1
In the current case n  17 . There were k  1 independent variables in the first regression and k  r  2
independent variables in the second regression. We have already found SST  1.71716 , SSR1  1.61734 ,
SSR2  1.6944 and SSE  0.0228 . So SSR 2 SSR1  1.6944  1.6173  0.0771 and our ANOVA follows.
Source
First Regression
2nd Regression
SS
1.6173
0.0771
DF
1
1
MS
Fcalc
1,14  4.60
47.32s F.05
0.0771
Error
0.0228
14
0.001629
Total
1.7172
16
We can get the same results using R 2 . Remember RY2.12  0.9867 and RY2.1  .9417 .
Source
SS
DF
MS
Fcalc
RY2.1  .9417
RY2.12  RY2.1  .9867  .9417  .0450
1  RY2.12  1  .9867  .0133
First Regression
2nd Regression
Error
F
F
1
1
.0450
14
..000950
1,14  4.60
47.37s F.05
Total
Column must add to 1.000
16
Note that these seem to show that the second independent variable made a significant improvement in the
amount of variation in Y explained by the independent variables. We have rejected our null hypothesis of
no improvement.
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]
Solution: We have the equation Yˆ  b0  b1 X 1  b2 X 2  0.4054  0.03097X 1  0.01242X 2 . For 2007 we
were given that the year was 28 and the Crude price was 90, so we can
Yˆ  0.4054  0.0309790  0.0124228  0.4054  2.7873 0.3478  3.54 . This looks a little closer.
s
We can find an approximate confidence interval  Y0  Yˆ0  t e and an approximate prediction interval
n
Y  Yˆ  t s . s is the square root of MSE. From the ANOVA in b) the mean square error is MSE =
0
0
e
e
14
 2.145 =2.145. So the
0.0016286 and its square root is .0404. Our degrees of freedom were 14 and t .025
0.0016286
 3.54  0.006 . The prediction interval, which is really the
17
only relevant interval here is 3.54  2.145 .0404   3.54  0.09 . I guess we can only wait and see.
confidence interval is 3.54  2.145
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)
All of these are available as an appendix A.
11
252y0771t 11/28/07
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
The solution that follows is for Version 0. All other versions will be covered in appendix B.
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)
Solution: This solution follows the material in 252anovaex3. New material is added in boldface. This
table is set up to do both one-way and two-way ANOVA. For One-way ANOVA use only the material
below the 3 columns and the relevant sums.
Venue
Fifi
Adelina
Maria
Sum
SS
ni
x i 
x
Milan
Venice
Naples
Rome
Sum
nj
36
22
19
16
93
4
39
14
20
18
91
4
21
32
28
22
103
4
96
68
67
56
287
12
x j 
23.25
22.75
25.75
(23.9167)
x
SS
2397
540.5625
2441
517.5625
2733
663.0625
7571
1721.1875
2
 xijk
x j  2
3
3
3
3
12
n
32.0000
22.6667
22.3333
18.6667
(23.9167)
x
3258
1704
1545
1064
7571
2
 xijk
and

x
x.2j
2
SSB 
 x i 2
 x  287
 7571 For the column means x.1  23 .25 , x.2  22.75 and x.3  25 .75 , which means
 1721 .1875 . (This sum is only useful if there are the same number of items in each column.) The
overall mean is x 
 x
.j
 x  287  23.9167 . We can now compute the sum of squares between treatments.
n
x
12
   n j x.2j  nx 2  423.25 2  422.752  425.75 2 1223.9167 2
2
 41721 .1875   1223 .9167 2  20 .6475 . There is a quite forgivable rounding error here, if we use x
unrounded we get 20.6667.
SST 
 x
2
 nx 2  7571 1223.91672  706.8975. Again, with an unrounded value of x , we get
706.9166.
So here is our ANOVA table. Because the computed F is smaller than the table F, we cannot reject H 0 .
Source
SS
DF
MS
F.10
H0
F
Between
Within
Total
20.6667
2
10.3334
686.2499
706.9166
9
11
76.2500
0.136ns
F 2,9   3.01
2
1024.00
513.78
498.78
348.44
2385.00
 x .2j .
To summarize the material above, we have for one-way ANOVA n  12 , n1  n2  n3  4 ,
i
Column means equal
12
252y0771t 11/28/07
b) Still assuming that these are random samples, use a method that compares medians instead. (3)
Solution: We must now use the Kruskal-Wallis test. The data is presented with rankings.
Row
1
2
3
4
Fifi
36
22
19
16
Rank Fifi
11.0
7.5
4.0
2.0
24.5
Adelina
39
14
20
18
Rank Adelina
12
1
5
3
21
Maria
21
32
28
22
Rank Maria
6.0
10.0
9.0
7.5
32.5
To check our totals, note that 24.5 + 21 + 32.5 = 78 and that the sum of the numbers 1 through 12 is
12 13 
 78 . No matter what the dimension of the problem, do not change the 12 in the following
2
formula. Of course, you must change n. Now, compute the Kruskal-Wallis statistic
2
2
2 
 12
 SRi 2 


  3n  1   12  24 .5  21  32 .5   313 
H 
4
4 
 nn  1 i  ni 
12 13   4


1
524 .375   39  1.3365 .
13
The 4,4,4 part of the K-W table follows. We can only conclude that the p-value is above .104. This value is
above any commonly used significance level, so we cannot reject our null hypothesis.
4
4
4
7.6538
7.5385
5.6923
5.6538
4.6539
4.5001
.008
.011
.049
.054
.097
.104
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 means. (5)
Solution: This solution follows the material in 252anovaex3. New material is added in boldface. This
table is set up to do both one-way and two-way ANOVA.
Venue
Fifi
Adelina
Maria
Sum
SS
ni
x i 
x
Milan
Venice
Naples
Rome
Sum
nj
x j 
SS
x j 
2
36
22
19
16
93
4
39
14
20
18
91
4
21
32
28
22
103
4
96
68
67
56
287
12
23.25
22.75
25.75
(23.9167)
x
7571
1721.1875
2
 xijk
2397
540.5625
2441
517.5625
2733
663.0625
3
3
3
3
12
n
32.0000
22.6667
22.3333
18.6667
(23.9167)
x
3258
1704
1545
1064
7571
2
 xijk
row means
 x  287
x
2
i
and
x
2
 x i 2
 x .2j .
 7571 . For the column means
 2835 . . The overall mean is x 
x
2
 j
 1721 .1875 and for the
 x  287  23.9167 . We can now compute the
n
12
sums of squares between rows and columns. Because there is only one item per cell, there in no interaction
sum of squares.
x
SSR  C  x
SSC  R
2
1024.00
513.78
498.78
348.44
2385.00
To summarize the material above, we have for two-way ANOVA n  12 . There are R  4 rows and
C  3 columns.
i
2
.j
 nx 2  41721 .1875   1223 .9167 2  20 .6475 . Actually 20.6667 with unrounded x .
2
i.
 nx 2  32385   1223 .9167 2  290 .8975 . Without rounding, this is closer to 290.9167
13
252y0771t 11/28/07
SST 
 x
2
 nx 2  7571 1223.91672  706.8975. Again, with an unrounded value of x , we get
706.9166.
Our ANOVA table is thus as shown below. In this version neither of the hypotheses is rejected. This is not
always true.
Source
SS
DF
MS
F
F
H0


3
,
6
290.9167
3
96.9722
1.47ns
Row
means
equal
Rows  A
F
 3.10
Columns B 
20.6667
2
10.3334
.10
2,6
F.10
0.16ns
 3.46
Column means equal
0
0
None
None
None
No Interaction
Interaction  AB 
Within
395.3341
6
65.8890
Total
706.9175
11
d) Do two different types of confidence intervals between Milan and the least enthusiastic opera house.
Explain the difference between the intervals. (2)
Solution: Though you might not have picked up on the error in c) before now. This should have alerted
you. Why were you all asleep? Since I blew the statement of the previous section, but no one complained,
credit will be given for any sets of meaningful comparison of rows and columns. The row means were as
below.
Milan
32.0000
Venice
22.6667
Naples
22.3333
Rome
18.6667
The least enthusiastic house seems to have been Rome. The difference between the means is
x1  x 4   32.0000 18.6667  13.3333 and MSW  65.8890. There are R  4 rows, C  3 columns and
P  1 measurements per cell. ( R  1)(C  1)  32  6 .
2 MSW

PC
265 .8890 
 43 .9260  6.6277
3
The intervals presented in the outline were as below.
Scheffé Confidence Interval
If we desire intervals that will simultaneously be valid for a given confidence level for all possible intervals
between means, use the following formulas.
For row means, use 1   2  x1  x 2  
R  1FR 1, RC P 1 2MSW
PC
For column means, use  1   2  x1  x2  
.
C  1FC 1, RCP 1 2MSW
PR
.
Note that if P  1 we replace RC P  1 with R  1C  1 .
The requested row mean contrast is
2MSW
 13 .3333  33.29 6.6277   13.3333  20.8220 . Note that
1   4  x1  x 4   3F3,6 
PC
this says that the difference is insignificant.
Bonferroni Confidence Interval with m  1 is an individual interval, not simultaneously valid at the given
confidence level.
For row means use 1   2  x1  x 2   t RC P 1
2m
2MSW
.
PC
For column means use  1   2  x1  x2   t RC P 1
2m
2MSW
.
PR
14
252y0771t 11/28/07
Note that if P  1 we replace RC P  1 with
R  1C  1 .
The requested row mean contrast is 1   4  x1  x 4   t 6  
2
2MSW
 13.3333  1.943 6.6277 
5
 13.3333  12.8776 . I’m astonished that this is significant.
Tukey Confidence Interval These are sort of a loose version of the Scheffé intervals.
For row means, use 1   2  x1  x 2   qR , RC P 1
MSW
.
PC
For column means, use  1   2  x1  x2   qC , RC P 1
Note that if P  1 , replace RC P  1 with
MSW
PR
R  1C  1 .
The requested row mean contrast is 1   4  x1  x 4   q4,6 
MSW
. We do not have tables for
3
  .10 . So these are not given.
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)
Solution: We must now use the Friedman test. The data is presented with rankings within the rows.
Venue
Milan
Venice
Naples
Rome
Fifi Rank Fifi
36
2
22
2
19
1
16
1
6
Adelina Rank Adelina
39
3
14
1
20
2
18
2
8
Maria Rank Maria
21
1
32
3
28
3
22
3
10
To check the ranking, note that the sum of the three rank sums is 6 + 8 + 10 = 24, and that if we have C
RC C  1  434  

columns and R rows, the sum of the column rank sums should be
  24 .
2
 2 
 12
Now compute the Friedman statistic  F2  
 rc c  1


 SR   3r c  1
2
i
i

 12
62  82  10 2   344   1 200   48  2 . The part of the Friedman table (Table 8)





4
3
4
4



for 3 columns and 4 rows is below.
c  3,
r 4
 F2
0.000
0.500
1.500
2.000
3.500
4.500
6.000
6.500
8.000
p  value
1.000
.931
.653
.431
.273
.125
.069
.042
.005
According to this table the p-value for  F2  2 is .431. Since the p-value is above   .10 , do not reject the
null hypothesis of equal medians. I’m shocked that Minitab, though it gets the same value of  F2 that I do,
gives it a different p-value. The only think that I can think of is that it is treating this as a regular  2 .
In any case we still do not believe that there is any difference between the sopranos.
15
252y0771t 11/28/07
f) Do you prefer the methods that compare medians or means? Don’t answer this unless you can
demonstrate an informed opinion. (1)
Solution: Caution would lead us to using a comparison of medians since we can have no proof that
applause follows the Normal distribution.
g) (Extra credit) Do a Levine test on these data and explain what it tests and shows.(3)
Solution: This test is quite simple. It can be used for non-Normal data and can be used to compare two
columns as well as more than two columns.
Row
1
2
3
4
Fifi
36
22
19
16
Adelina
39
14
20
18
Maria
21
32
28
22
(i) Find the median of each column. (This is the middle number or the average of the two middle numbers.)
Row Fifi Adelina
1
36
39
2
22
14
3
19
20
4
16
18
Median 20.5
19.0
Maria
21
32
28
22
25.0
(ii) Subtract the median of each column from the column from which it comes.
Row
1
2
3
4
.
Fifi
15.5
1.5
-1.5
-4.5
Adelina
20
-5
1
-1
Take the absolute value of the result.
Row
1
2
3
4
Fifi Adelina
15.5
20
1.5
5
1.5
1
4.5
1
Maria
-4
7
3
-3
Maria
4
7
3
3
(iii) Do a 1-way ANOVA on the result. If the results would lead you to reject the null hypothesis (because
the computed F is above the table F or the p-value is below your significance level), reject the null
hypothesis of equal variances.
Here is our ANOVA table. Because the computed F is smaller than the table F, we cannot reject H 0 .
Source
SS
DF
MS
F.10
H0
F
Between
Within
Total
12.7
2
388.3
400.9
9
11
6.35
0.15ns
F 2,9   3.01
Equal Variances
43.144
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.
See appendix B for my versions.
16
252y0771t 11/28/07
Appendix A Regression Problem
————— 11/28/2007 8:04:22 PM ————————————————————
Welcome to Minitab, press F1 for help.
MTB > WOpen "C:\Documents and Settings\rbove\My Documents\Minitab\252x077110a.MTW".
Retrieving worksheet from file: 'C:\Documents and Settings\rbove\My
Documents\Minitab\252x0771-10a.MTW'
Worksheet was saved on Wed Nov 28 2007
Version 0
Results for: 252x0771-10a.MTW
MTB > Execute "C:\Documents and Settings\RBOVE\My
Documents\Minitab\SpareParts.mtb" 1.
Executing from file: C:\Documents and Settings\RBOVE\My
Documents\Minitab\SpareParts.mtb
Data Display
Row
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
GP0_1
1.25
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
CP0_1
26.07
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
Yr0_1
1
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
Ysq
1.5625
1.3456
1.2996
1.2769
1.2321
1.2321
1.3225
1.5129
1.5129
1.1236
1.3689
2.2801
2.1316
1.8496
2.5281
3.5344
5.2900
X1sq
679.64
493.73
363.28
339.66
269.29
243.05
296.87
428.90
362.52
156.75
306.60
798.63
526.70
580.81
813.96
1367.52
2523.05
X2sq
1
121
144
169
196
225
256
289
324
361
400
441
484
529
576
625
676
X1Y
32.588
25.775
21.728
20.826
18.215
17.305
19.815
25.473
23.419
13.271
20.487
42.673
33.507
32.776
45.363
69.522
115.529
X2Y
1.25
12.76
13.68
14.69
15.54
16.65
18.40
20.91
22.14
20.14
23.40
31.71
32.12
31.28
38.16
47.00
59.80
X1X2
26.07
244.42
228.72
239.59
229.74
233.85
275.68
352.07
342.72
237.88
350.20
593.46
504.90
554.30
684.72
924.50
1305.98
Data Display
sumY
sumX1
sumX2
sumYsq
sumX1sq
sumX2sq
sumX1y
sumX2y
sumx1x2
22.8400
395.840
297.000
32.4034
10551.0
5817.00
578.272
419.630
7328.80
Data Display
meanY
meanX1
meanX2
SSX1
SSX2
SSY
SX1Y
SX2Y
SX1X2
n
1.34353
23.2847
17.4706
1333.96
628.235
1.71719
46.4489
20.6018
413.242
17.0000
17
252y0771t 11/28/07
Regression Analysis: GP0_1 versus CP0_1
The regression equation is
GP0_1 = 0.533 + 0.0348 CP0_1
Predictor
Coef
SE Coef
Constant
0.53275
0.05565
CP0_1
0.034820 0.002234
S = 0.0815786
R-Sq = 94.2%
Analysis of Variance
Source
DF
SS
Regression
1 1.6174
Residual Error 15 0.0998
Total
16 1.7172
T
P
9.57 0.000
15.59 0.000
R-Sq(adj) = 93.8%
MS
1.6174
0.0067
F
243.03
P
0.000
Unusual Observations
Obs CP0_1
GP0_1
Fit SE Fit Residual St Resid
1
26.1 1.2500 1.4405 0.0207
-0.1905
-2.41R
17
50.2 2.3000 2.2818 0.0634
0.0182
0.35 X
R denotes an observation with a large standardized residual.
X denotes an observation whose X value gives it large influence.
Regression Analysis: GP0_1 versus Yr0_1
The regression equation is
GP0_1 = 0.771 + 0.0328 Yr0_1
Predictor
Coef SE Coef
Constant
0.7706
0.1945
Yr0_1
0.03279 0.01051
S = 0.263514
R-Sq = 39.3%
T
P
3.96 0.001
3.12 0.007
R-Sq(adj) = 35.3%
Analysis of Variance
Source
DF
SS
MS
F
P
Regression
1 0.67560 0.67560 9.73 0.007
Residual Error 15 1.04159 0.06944
Total
16 1.71719
Unusual Observations
Obs Yr0_1
GP0_1
Fit SE Fit Residual St Resid
1
1.0 1.2500 0.8034 0.1846
0.4466
2.37RX
17
26.0 2.3000 1.6232 0.1101
0.6768
2.83R
R denotes an observation with a large standardized residual.
X denotes an observation whose X value gives it large influence.
Regression Analysis: GP0_1 versus CP0_1, Yr0_1
The regression equation is
GP0_1 = 0.405 + 0.0310 CP0_1 + 0.0124 Yr0_1
Predictor
Coef
SE Coef
Constant
0.40536
0.03307
CP0_1
0.030973 0.001235
Yr0_1
0.012420 0.001799
S = 0.0402384
R-Sq = 98.7%
Analysis of Variance
Source
DF
SS
Regression
2 1.69452
Residual Error 14 0.02267
Total
16 1.71719
Source
CP0_1
Yr0_1
DF
1
1
T
P
12.26 0.000
25.09 0.000
6.90 0.000
R-Sq(adj) = 98.5%
MS
0.84726
0.00162
F
523.28
P
0.000
Seq SS
1.61736
0.07716
Unusual Observations
Obs CP0_1
GP0_1
Fit
1
26.1 1.25000 1.22524
SE Fit
0.03282
Residual
0.02476
St Resid
1.06 X
18
252y0771t 11/28/07
14
24.1 1.36000 1.43745 0.01364 -0.07745
-2.05R
17
50.2 2.30000 2.28403 0.03125
0.01597
0.63 X
R denotes an observation with a large standardized residual.
X denotes an observation whose X value gives it large leverage.
19
252y0771t 11/28/07
MTB > WOpen "C:\Documents and Settings\rbove\My Documents\Minitab\252x077111a.MTW".
Retrieving worksheet from file: 'C:\Documents and Settings\rbove\My
Documents\Minitab\252x0771-11a.MTW'
Worksheet was saved on Wed Nov 28 2007
Version 1
Results for: 252x0771-11a.MTW
MTB > Execute "C:\Documents and Settings\RBOVE\My
Documents\Minitab\SpareParts.mtb" 1.
Executing from file: C:\Documents and Settings\RBOVE\My
Documents\Minitab\SpareParts.mtb
Data Display
Row
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
GP1_1
1.38
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
CP1_1
35.24
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
Yr1_1
2
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
Ysq
1.9044
1.3456
1.2996
1.2769
1.2321
1.2321
1.3225
1.5129
1.5129
1.1236
1.3689
2.2801
2.1316
1.8496
2.5281
3.5344
5.2900
X1sq
1241.86
493.73
363.28
339.66
269.29
243.05
296.87
428.90
362.52
156.75
306.60
798.63
526.70
580.81
813.96
1367.52
2523.05
Data Display
Sums of columns above.
Data Display
Spare Parts computation
sumY
sumX1
sumX2
sumYsq
sumX1sq
sumX2sq
sumX1y
sumX2y
sumx1x2
meanY
meanX1
meanX2
SSX1
SSX2
SSY
SX1Y
SX2Y
SX1X2
n
X2sq
4
121
144
169
196
225
256
289
324
361
400
441
484
529
576
625
676
X1Y
48.631
25.775
21.728
20.826
18.215
17.305
19.815
25.473
23.419
13.271
20.487
42.673
33.507
32.776
45.363
69.522
115.529
X2Y
2.76
12.76
13.68
14.69
15.54
16.65
18.40
20.91
22.14
20.14
23.40
31.71
32.12
31.28
38.16
47.00
59.80
X1X2
70.48
244.42
228.72
239.59
229.74
233.85
275.68
352.07
342.72
237.88
350.20
593.46
504.90
554.30
684.72
924.50
1305.98
22.9700
405.010
298.000
32.7453
11113.2
5820.00
594.315
421.140
7373.21
1.35118
23.8241
17.5294
1464.19
596.235
1.70878
47.0753
18.4894
273.623
17.0000
Regression Analysis: GP1_1 versus CP1_1 Note that p-values are very low implying that
coefficients are significant.
The regression equation is
GP1_1 = 0.585 + 0.0322 CP1_1
Predictor
Coef
SE Coef
T
P
Constant
0.58520
0.07623
7.68 0.000
CP1_1
0.032151 0.002982 10.78 0.000
S = 0.114091
R-Sq = 88.6%
R-Sq(adj) = 87.8%
20
252y0771t 11/28/07
Analysis of Variance
A low p-value means to reject hypothesis that there is no relation between
dependent variables and independent variables.
Source
DF
SS
MS
F
P
Regression
1 1.5135 1.5135 116.28 0.000
Residual Error 15 0.1953 0.0130
Total
16 1.7088
Unusual Observations
Obs CP1_1
GP1_1
Fit SE Fit Residual St Resid
1
35.2 1.3800 1.7182 0.0439
-0.3382
-3.21R
17
50.2 2.3000 2.2002 0.0835
0.0998
1.28 X
R denotes an observation with a large standardized residual.
X denotes an observation whose X value gives it large influence.
Regression Analysis: GP1_1 versus Yr1_1 Note that p-values are below 5% implying that
coefficients are significant at 5% level.
The regression equation is
GP1_1 = 0.808 + 0.0310 Yr1_1
Predictor
Coef SE Coef
Constant
0.8076
0.2085
Yr1_1
0.03101 0.01127
S = 0.275126
R-Sq = 33.6%
T
P
3.87 0.001
2.75 0.015
R-Sq(adj) = 29.1%
Analysis of Variance A p-value below 5% means to reject hypothesis that there is no relation between
dependent variables and independent variables at 5% level.
Source
DF
SS
MS
F
P
Regression
1 0.57336 0.57336 7.57 0.015
Residual Error 15 1.13541 0.07569
Total
16 1.70878
Unusual Observations
Obs Yr1_1
GP1_1
Fit SE Fit Residual St Resid
1
2.0 1.3800 0.8696 0.1873
0.5104
2.53RX
17
26.0 2.3000 1.6139 0.1165
0.6861
2.75R
R denotes an observation with a large standardized residual.
X denotes an observation whose X value gives it large influence.
MTB > regress c1 2 c2 c3
Regression Analysis: GP1_1 versus CP1_1, Yr1_1 Note that p-values are very low implying
that coefficients are significant.
The regression equation is
GP1_1 = 0.353 + 0.0288 CP1_1 + 0.0178 Yr1_1
Predictor
Coef
SE Coef
Constant
0.35269
0.03528
CP1_1
0.028828 0.001106
Yr1_1
0.017780 0.001733
S = 0.0404630
R-Sq = 98.7%
T
P
10.00 0.000
26.07 0.000
10.26 0.000
R-Sq(adj) = 98.5%
Analysis of Variance
A low p-value means to reject hypothesis that there is no relation between
dependent variables and independent variables.
Source
DF
SS
MS
F
P
Regression
2 1.68585 0.84293 514.84 0.000
Residual Error 14 0.02292 0.00164
Total
16 1.70878
Source
CP1_1
Yr1_1
DF
1
1
Seq SS
1.51353
0.17233
Unusual Observations
21
252y0771t 11/28/07
Obs CP1_1
GP1_1
Fit
SE Fit Residual St Resid
1
35.2 1.38000 1.40416 0.03434 -0.02416
-1.13 X
14
24.1 1.36000 1.45640 0.01359 -0.09640
-2.53R
17
50.2 2.30000 2.26303 0.03023
0.03697
1.37 X
R denotes an observation with a large standardized residual.
X denotes an observation whose X value gives it large influence.
22
252y0771t 11/28/07
MTB > WOpen "C:\Documents and Settings\rbove\My Documents\Minitab\252x077112a.MTW".
Retrieving worksheet from file: 'C:\Documents and Settings\rbove\My
Documents\Minitab\252x0771-12a.MTW'
Worksheet was saved on Wed Nov 28 2007
Version 2
Results for: 252x0771-12a.MTW
MTB > Execute "C:\Documents and Settings\RBOVE\My
Documents\Minitab\SpareParts.mtb" 1.
Executing from file: C:\Documents and Settings\RBOVE\My
Documents\Minitab\SpareParts.mtb
Data Display
Row
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
GP2_1
1.30
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
CP2_1
31.87
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
Yr2_1
3
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
Ysq
1.6900
1.3456
1.2996
1.2769
1.2321
1.2321
1.3225
1.5129
1.5129
1.1236
1.3689
2.2801
2.1316
1.8496
2.5281
3.5344
5.2900
X1sq
1015.70
493.73
363.28
339.66
269.29
243.05
296.87
428.90
362.52
156.75
306.60
798.63
526.70
580.81
813.96
1367.52
2523.05
Data Display
Sums of columns above.
Data Display
Spare Parts computation
sumY
sumX1
sumX2
sumYsq
sumX1sq
sumX2sq
sumX1y
sumX2y
sumx1x2
meanY
meanX1
meanX2
SSX1
SSX2
SSY
SX1Y
SX2Y
SX1X2
n
X2sq
9
121
144
169
196
225
256
289
324
361
400
441
484
529
576
625
676
X1Y
41.431
25.775
21.728
20.826
18.215
17.305
19.815
25.473
23.419
13.271
20.487
42.673
33.507
32.776
45.363
69.522
115.529
X2Y
3.90
12.76
13.68
14.69
15.54
16.65
18.40
20.91
22.14
20.14
23.40
31.71
32.12
31.28
38.16
47.00
59.80
X1X2
95.61
244.42
228.72
239.59
229.74
233.85
275.68
352.07
342.72
237.88
350.20
593.46
504.90
554.30
684.72
924.50
1305.98
22.8900
401.640
299.000
32.5309
10887.0
5825.00
587.115
422.280
7398.34
1.34647
23.6259
17.5882
1397.93
566.118
1.71019
46.3187
19.6853
334.201
17.0000
Regression Analysis: GP2_1 versus CP2_1 Note that p-values are very low implying that
coefficients are significant.
The regression equation is
GP2_1 = 0.564 + 0.0331 CP2_1
Predictor
Coef
SE Coef
T
P
Constant
0.56366
0.07321
7.70 0.000
CP2_1
0.033134 0.002893 11.45 0.000
S = 0.108161
R-Sq = 89.7%
R-Sq(adj) = 89.1%
23
252y0771t 11/28/07
Analysis of Variance
A low p-value means to reject hypothesis that there is no relation between
dependent variables and independent variables.
Source
DF
SS
MS
F
P
Regression
1 1.5347 1.5347 131.18 0.000
Residual Error 15 0.1755 0.0117
Total
16 1.7102
Unusual Observations
Obs CP2_1
GP2_1
Fit SE Fit Residual St Resid
1
31.9 1.3000 1.6196 0.0355
-0.3196
-3.13R
17
50.2 2.3000 2.2280 0.0813
0.0720
1.01 X
R denotes an observation with a large standardized residual.
X denotes an observation whose X value gives it large influence.
Regression Analysis: GP2_1 versus Yr2_1 Note that p-values are very low implying that
coefficients are significant.
The regression equation is
GP2_1 = 0.735 + 0.0348 Yr2_1
Predictor
Coef SE Coef
Constant
0.7349
0.2034
Yr2_1
0.03477 0.01099
S = 0.261493
R-Sq = 40.0%
T
P
3.61 0.003
3.16 0.006
R-Sq(adj) = 36.0%
Analysis of Variance
A low p-value means to reject hypothesis that there is no relation between
dependent variables and independent variables.
Source
DF
SS
MS
F
P
Regression
1 0.68451 0.68451 10.01 0.006
Residual Error 15 1.02568 0.06838
Total
16 1.71019
Unusual Observations
Obs Yr2_1
GP2_1
Fit SE Fit Residual St Resid
1
3.0 1.3000 0.8392 0.1724
0.4608
2.34RX
17
26.0 2.3000 1.6390 0.1121
0.6610
2.80R
R denotes an observation with a large standardized residual.
X denotes an observation whose X value gives it large influence.
MTB > regress c1 2 c2 c3
Regression Analysis: GP2_1 versus CP2_1, Yr2_1 Note that p-values are very low implying
that coefficients are significant.
The regression equation is
GP2_1 = 0.352 + 0.0289 CP2_1 + 0.0177 Yr2_1
Predictor
Coef
SE Coef
Constant
0.35218
0.03509
CP2_1
0.028899 0.001168
Yr2_1
0.017712 0.001836
S = 0.0404845
R-Sq = 98.7%
T
P
10.04 0.000
24.73 0.000
9.65 0.000
R-Sq(adj) = 98.5%
Analysis of Variance
A low p-value means to reject hypothesis that there is no relation between
dependent variables and independent variables.
Source
DF
SS
MS
F
P
Regression
2 1.68724 0.84362 514.72 0.000
Residual Error 14 0.02295 0.00164
Total
16 1.71019
Source
CP2_1
Yr2_1
DF
1
1
Seq SS
1.53471
0.15254
24
252y0771t 11/28/07
Unusual Observations
Obs CP2_1
GP2_1
Fit
SE Fit Residual St Resid
1
31.9 1.30000 1.32633 0.03317 -0.02633
-1.13 X
14
24.1 1.36000 1.45603 0.01383 -0.09603
-2.52R
17
50.2 2.30000 2.26430 0.03067
0.03570
1.35 X
R denotes an observation with a large standardized residual.
X denotes an observation whose X value gives it large influence.
25
252y0771t 11/28/07
MTB > WOpen "C:\Documents and Settings\rbove\My Documents\Minitab\252x077113a.MTW".
Retrieving worksheet from file: 'C:\Documents and Settings\rbove\My
Documents\Minitab\252x0771-13a.MTW'
Worksheet was saved on Wed Nov 28 2007
Version 3
Results for: 252x0771-13a.MTW
MTB > Execute "C:\Documents and Settings\RBOVE\My
Documents\Minitab\SpareParts.mtb" 1.
Executing from file: C:\Documents and Settings\RBOVE\My
Documents\Minitab\SpareParts.mtb
Data Display
Row
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
GP3_1
1.24
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
CP3_1
26.99
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
YR3_1
4
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
Ysq
1.5376
1.3456
1.2996
1.2769
1.2321
1.2321
1.3225
1.5129
1.5129
1.1236
1.3689
2.2801
2.1316
1.8496
2.5281
3.5344
5.2900
X1sq
728.46
493.73
363.28
339.66
269.29
243.05
296.87
428.90
362.52
156.75
306.60
798.63
526.70
580.81
813.96
1367.52
2523.05
Data Display
Sums of columns above.
Data Display
Spare Parts computation
sumY
sumX1
sumX2
sumYsq
sumX1sq
sumX2sq
sumX1y
sumX2y
sumx1x2
meanY
meanX1
meanX2
SSX1
SSX2
SSY
SX1Y
SX2Y
SX1X2
n
X2sq
16
121
144
169
196
225
256
289
324
361
400
441
484
529
576
625
676
X1Y
33.468
25.775
21.728
20.826
18.215
17.305
19.815
25.473
23.419
13.271
20.487
42.673
33.507
32.776
45.363
69.522
115.529
X2Y
4.96
12.76
13.68
14.69
15.54
16.65
18.40
20.91
22.14
20.14
23.40
31.71
32.12
31.28
38.16
47.00
59.80
X1X2
107.96
244.42
228.72
239.59
229.74
233.85
275.68
352.07
342.72
237.88
350.20
593.46
504.90
554.30
684.72
924.50
1305.98
22.8300
396.760
300.000
32.3785
10599.8
5832.00
579.152
423.340
7410.69
1.34294
23.3388
17.6471
1339.88
537.882
1.71915
46.3264
20.4576
409.043
17.0000
Regression Analysis: GP3_1 versus CP3_1 Note that p-values are very low implying that
coefficients are significant.
The regression equation is
GP3_1 = 0.536 + 0.0346 CP3_1
Predictor
Coef
SE Coef
Constant
0.53600
0.06036
CP3_1
0.034575 0.002417
S = 0.0884778
R-Sq = 93.2%
T
P
8.88 0.000
14.30 0.000
R-Sq(adj) = 92.7%
26
252y0771t 11/28/07
Analysis of Variance
A low p-value means to reject hypothesis that there is no relation between
dependent variables and independent variables.
Source
DF
SS
MS
F
P
Regression
1 1.6017 1.6017 204.61 0.000
Residual Error 15 0.1174 0.0078
Total
16 1.7192
Unusual Observations
Obs CP3_1
GP3_1
Fit SE Fit Residual St Resid
1
27.0 1.2400 1.4692 0.0232
-0.2292
-2.68R
17
50.2 2.3000 2.2727 0.0685
0.0273
0.49 X
R denotes an observation with a large standardized residual.
X denotes an observation whose X value gives it large influence.
Regression Analysis: GP3_1 versus YR3_1
Note that p-values are very low implying that
coefficients are significant.
The regression equation is
GP3_1 = 0.672 + 0.0380 YR3_1
Predictor
Coef SE Coef
Constant
0.6718
0.2000
YR4_1
0.03803 0.01080
S = 0.250476
R-Sq = 45.3%
T
P
3.36 0.004
3.52 0.003
R-Sq(adj) = 41.6%
Analysis of Variance
A low p-value means to reject hypothesis that there is no relation
between dependent variables and independent variables.
Source
DF
SS
MS
F
P
Regression
1 0.77808 0.77808 12.40 0.003
Residual Error 15 0.94107 0.06274
Total
16 1.71915
Unusual Observations
Obs YR4_1
GP3_1
Fit SE Fit Residual St Resid
1
4.0 1.2400 0.8239 0.1594
0.4161
2.15RX
17
26.0 2.3000 1.6606 0.1088
0.6394
2.83R
R denotes an observation with a large standardized residual.
X denotes an observation whose X value gives it large influence.
MTB > regress c1 2 c2 c3
Regression Analysis: GP3_1 versus CP3_1, YR3_1 Note that p-values are very low implying
that coefficients are significant.
The regression equation is
GP3_1 = 0.375 + 0.0299 CP3_1 + 0.0153 YR3_1
Predictor
Coef
SE Coef
Constant
0.37512
0.03306
CP3_1
0.029907 0.001204
YR4_1
0.015290 0.001900
S = 0.0386059
R-Sq = 98.8%
T
P
11.35 0.000
24.85 0.000
8.05 0.000
R-Sq(adj) = 98.6%
Analysis of Variance
A low p-value means to reject hypothesis that there is no relation between
dependent variables and independent variables.
Source
DF
SS
MS
F
P
Regression
2 1.69829 0.84914 569.74 0.000
Residual Error 14 0.02087 0.00149
Total
16 1.71915
Source
CP3_1
YR4_1
DF
1
1
Seq SS
1.60173
0.09656
27
252y0771t 11/28/07
Unusual Observations
Obs CP3_1
GP3_1
Fit
SE Fit Residual St Resid
1
27.0 1.24000 1.24347 0.02981 -0.00347
-0.14 X
14
24.1 1.36000 1.44755 0.01353 -0.08755
-2.42R
17
50.2 2.30000 2.27490 0.02987
0.02510
1.03 X
R denotes an observation with a large standardized residual.
X denotes an observation whose X value gives it large influence.
28
252y0771t 11/28/07
MTB > WOpen "C:\Documents and Settings\rbove\My Documents\Minitab\252x077114a.MTW".
Retrieving worksheet from file: 'C:\Documents and Settings\rbove\My
Documents\Minitab\252x0771-14a.MTW'
Worksheet was saved on Wed Nov 28 2007
Version 4
Results for: 252x0771-14a.MTW
MTB > Execute "C:\Documents and Settings\RBOVE\My
Documents\Minitab\SpareParts.mtb" 1.
Executing from file: C:\Documents and Settings\RBOVE\My
Documents\Minitab\SpareParts.mtb
Data Display
Row
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
GP4_1
1.21
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
CP4_1
28.63
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
Yr4_1
5
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
Ysq
1.4641
1.3456
1.2996
1.2769
1.2321
1.2321
1.3225
1.5129
1.5129
1.1236
1.3689
2.2801
2.1316
1.8496
2.5281
3.5344
5.2900
X1sq
819.68
493.73
363.28
339.66
269.29
243.05
296.87
428.90
362.52
156.75
306.60
798.63
526.70
580.81
813.96
1367.52
2523.05
Data Display
Sums of columns above.
Data Display
Spare Parts computation
sumY
sumX1
sumX2
sumYsq
sumX1sq
sumX2sq
sumX1y
sumX2y
sumx1x2
meanY
meanX1
meanX2
SSX1
SSX2
SSY
SX1Y
SX2Y
SX1X2
n
X2sq
25
121
144
169
196
225
256
289
324
361
400
441
484
529
576
625
676
X1Y
34.642
25.775
21.728
20.826
18.215
17.305
19.815
25.473
23.419
13.271
20.487
42.673
33.507
32.776
45.363
69.522
115.529
X2Y
6.05
12.76
13.68
14.69
15.54
16.65
18.40
20.91
22.14
20.14
23.40
31.71
32.12
31.28
38.16
47.00
59.80
X1X2
143.15
244.42
228.72
239.59
229.74
233.85
275.68
352.07
342.72
237.88
350.20
593.46
504.90
554.30
684.72
924.50
1305.98
22.8000
398.400
301.000
32.3050
10691.0
5841.00
580.326
424.430
7445.88
1.34118
23.4353
17.7059
1354.39
511.529
1.72618
46.0017
20.7359
391.856
17.0000
Regression Analysis: GP4_1 versus CP4_1
Note that p-values are very low implying that
coefficients are significant.
The regression equation is
GP4_1 = 0.545 + 0.0340 CP4_1
Predictor
Coef
SE Coef
T
P
Constant
0.54520
0.07119
7.66 0.000
CP4_1
0.033965 0.002839 11.96 0.000
S = 0.104479
R-Sq = 90.5%
R-Sq(adj) = 89.9%
29
252y0771t 11/28/07
Analysis of Variance
A low p-value means to reject hypothesis that there is no relation
between dependent variables and independent variables.
Source
DF
SS
MS
F
P
Regression
1 1.5624 1.5624 143.14 0.000
Residual Error 15 0.1637 0.0109
Total
16 1.7262
Unusual Observations
Obs CP4_1
GP4_1
Fit SE Fit Residual St Resid
1
28.6 1.2100 1.5176 0.0293
-0.3076
-3.07R
17
50.2 2.3000 2.2513 0.0802
0.0487
0.73 X
R denotes an observation with a large standardized residual.
X denotes an observation whose X value gives it large influence.
Regression Analysis: GP4_1 versus Yr4_1 Note that p-values are very low implying that
coefficients are significant.
The regression equation is
GP4_1 = 0.623 + 0.0405 Yr4_1
Predictor
Coef SE Coef
Constant
0.6234
0.1991
Yr4_1
0.04054 0.01074
S = 0.242982
R-Sq = 48.7%
T
P
3.13 0.007
3.77 0.002
R-Sq(adj) = 45.3%
Analysis of Variance
A low p-value means to reject hypothesis that there is no relation between
dependent variables and independent variables.
Source
DF
SS
MS
F
P
Regression
1 0.84057 0.84057 14.24 0.002
Residual Error 15 0.88561 0.05904
Total
16 1.72618
Unusual Observations
Obs Yr4_1
GP4_1
Fit SE Fit Residual St Resid
1
5.0 1.2100 0.8261 0.1487
0.3839
2.00 X
17
26.0 2.3000 1.6774 0.1068
0.6226
2.85R
R denotes an observation with a large standardized residual.
X denotes an observation whose X value gives it large influence.
MTB > regress c1 2 c2 c3
Regression Analysis: GP4_1 versus CP4_1, Yr4_1
Note that p-values are very low implying
that coefficients are significant.
The regression equation is
GP4_1 = 0.341 + 0.0286 CP4_1 + 0.0187 Yr4_1
Predictor
Coef
SE Coef
Constant
0.34141
0.03710
CP4_1
0.028568 0.001307
Yr4_1
0.018652 0.002127
S = 0.0424387
R-Sq = 98.5%
T
P
9.20 0.000
21.86 0.000
8.77 0.000
R-Sq(adj) = 98.3%
Analysis of Variance
A low p-value means to reject hypothesis that there is no relation
between dependent variables and independent variables.
Source
DF
SS
MS
F
P
Regression
2 1.70096 0.85048 472.22 0.000
Residual Error 14 0.02521 0.00180
Total
16 1.72618
Source
CP4_1
Yr4_1
DF
1
1
Seq SS
1.56244
0.13852
30
252y0771t 11/28/07
Unusual Observations
Obs CP4_1
GP4_1
Fit SE Fit Residual St Resid
1
28.6 1.2100 1.2526 0.0325
-0.0426
-1.56 X
14
24.1 1.3600 1.4589 0.0150
-0.0989
-2.49R
17
50.2 2.3000 2.2614 0.0326
0.0386
1.42 X
R denotes an observation with a large standardized residual.
X denotes an observation whose X value gives it large influence.
31
252y0771t 11/28/07
MTB > WOpen "C:\Documents and Settings\rbove\My Documents\Minitab\252x077115a.MTW".
Retrieving worksheet from file: 'C:\Documents and Settings\rbove\My
Documents\Minitab\252x0771-15a.MTW'
Worksheet was saved on Wed Nov 28 2007
Version 5
Results for: 252x0771-15a.MTW
MTB > Execute "C:\Documents and Settings\RBOVE\My
Documents\Minitab\SpareParts.mtb" 1.
Executing from file: C:\Documents and Settings\RBOVE\My
Documents\Minitab\SpareParts.mtb
Data Display
Row
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
GP5_1
1.20
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
CP5_1
26.25
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
Yr5_1
6
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
Ysq
1.4400
1.3456
1.2996
1.2769
1.2321
1.2321
1.3225
1.5129
1.5129
1.1236
1.3689
2.2801
2.1316
1.8496
2.5281
3.5344
5.2900
X1sq
689.06
493.73
363.28
339.66
269.29
243.05
296.87
428.90
362.52
156.75
306.60
798.63
526.70
580.81
813.96
1367.52
2523.05
Data Display
Sums of columns above.
Data Display
Spare Parts computation
sumY
sumX1
sumX2
sumYsq
sumX1sq
sumX2sq
sumX1y
sumX2y
sumx1x2
meanY
meanX1
meanX2
SSX1
SSX2
SSY
SX1Y
SX2Y
SX1X2
n
X2sq
36
121
144
169
196
225
256
289
324
361
400
441
484
529
576
625
676
X1Y
31.500
25.775
21.728
20.826
18.215
17.305
19.815
25.473
23.419
13.271
20.487
42.673
33.507
32.776
45.363
69.522
115.529
X2Y
7.20
12.76
13.68
14.69
15.54
16.65
18.40
20.91
22.14
20.14
23.40
31.71
32.12
31.28
38.16
47.00
59.80
X1X2
157.50
244.42
228.72
239.59
229.74
233.85
275.68
352.07
342.72
237.88
350.20
593.46
504.90
554.30
684.72
924.50
1305.98
22.7900
396.020
302.000
32.2809
10560.4
5852.00
577.184
425.580
7460.23
1.34059
23.2953
17.7647
1335.00
487.059
1.72889
46.2843
20.7224
425.051
17.0000
Regression Analysis: GP5_1 versus CP5_1
Note that p-values are very low implying that
coefficients are significant.
The regression equation is
GP5_1 = 0.533 + 0.0347 CP5_1
Predictor
Coef
SE Coef
Constant
0.53294
0.06208
CP5_1
0.034670 0.002491
S = 0.0910001
R-Sq = 92.8%
T
P
8.59 0.000
13.92 0.000
R-Sq(adj) = 92.3%
32
252y0771t 11/28/07
Analysis of Variance
A low p-value means to reject hypothesis that there is no relation between
dependent variables and independent variables.
Source
DF
SS
MS
F
P
Regression
1 1.6047 1.6047 193.78 0.000
Residual Error 15 0.1242 0.0083
Total
16 1.7289
Unusual Observations
Obs CP5_1
GP5_1
Fit SE Fit Residual St Resid
1
26.3 1.2000 1.4430 0.0233
-0.2430
-2.76R
17
50.2 2.3000 2.2744 0.0706
0.0256
0.45 X
R denotes an observation with a large standardized residual.
X denotes an observation whose X value gives it large influence.
Regression Analysis: GP5_1 versus Yr5_1
Note that p-values are very low implying that
coefficients are significant.
The regression equation is
GP5_1 = 0.585 + 0.0425 Yr5_1
Predictor
Coef SE Coef
Constant
0.5848
0.1998
Yr5_1
0.04255 0.01077
S = 0.237661
R-Sq = 51.0%
T
P
2.93 0.010
3.95 0.001
R-Sq(adj) = 47.7%
Analysis of Variance
A low p-value means to reject hypothesis that there is no relation
between dependent variables and independent variables.
Source
DF
SS
MS
F
P
Regression
1 0.88165 0.88165 15.61 0.001
Residual Error 15 0.84724 0.05648
Total
16 1.72889
Unusual Observations
Obs Yr5_1
GP5_1
Fit SE Fit Residual St Resid
17
26.0 2.3000 1.6910 0.1058
0.6090
2.86R
R denotes an observation with a large standardized residual.
Regression Analysis: GP5_1 versus CP5_1, Yr5_1
Note that p-values are very low implying
that coefficients are significant.
The regression equation is
GP5_1 = 0.357 + 0.0293 CP5_1 + 0.0170 Yr5_1
Predictor
Coef
SE Coef
Constant
0.35684
0.03505
CP5_1
0.029251 0.001287
Yr5_1
0.017018 0.002130
S = 0.0399518
R-Sq = 98.7%
Analysis of Variance
Source
DF
SS
Regression
2 1.70655
Residual Error 14 0.02235
Total
16 1.72889
Source
CP5_1
Yr5_1
DF
1
1
T
P
10.18 0.000
22.73 0.000
7.99 0.000
R-Sq(adj) = 98.5%
MS
0.85327
0.00160
F
534.58
P
0.000
Seq SS
1.60468
0.10187
Unusual Observations
Obs CP5_1
GP5_1
Fit
SE Fit Residual St Resid
14
24.1 1.36000 1.45322 0.01439 -0.09322
-2.50R
17
50.2 2.30000 2.26862 0.03101
0.03138
1.25 X
R denotes an observation with a large standardized residual.
X denotes an observation whose X value gives it large influence.
33
252y0771t 11/28/07
MTB > WOpen "C:\Documents and Settings\rbove\My Documents\Minitab\252x077116a.MTW".
Retrieving worksheet from file: 'C:\Documents and Settings\rbove\My
Documents\Minitab\252x0771-16a.MTW'
Worksheet was saved on Wed Nov 28 2007
Version 6
Results for: 252x0771-16a.MTW
MTB > Execute "C:\Documents and Settings\RBOVE\My
Documents\Minitab\SpareParts.mtb" 1.
Executing from file: C:\Documents and Settings\RBOVE\My
Documents\Minitab\SpareParts.mtb
Data Display
Row
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
GP6_1
0.93
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
CP6_1
14.55
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
Yr6_1
7
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
Ysq
0.8649
1.3456
1.2996
1.2769
1.2321
1.2321
1.3225
1.5129
1.5129
1.1236
1.3689
2.2801
2.1316
1.8496
2.5281
3.5344
5.2900
X1sq
211.70
493.73
363.28
339.66
269.29
243.05
296.87
428.90
362.52
156.75
306.60
798.63
526.70
580.81
813.96
1367.52
2523.05
Data Display
Sums of columns above.
Data Display
Spare Parts computation
sumY
sumX1
sumX2
sumYsq
sumX1sq
sumX2sq
sumX1y
sumX2y
sumx1x2
meanY
meanX1
meanX2
SSX1
SSX2
SSY
SX1Y
SX2Y
SX1X2
n
X2sq
49
121
144
169
196
225
256
289
324
361
400
441
484
529
576
625
676
X1Y
13.532
25.775
21.728
20.826
18.215
17.305
19.815
25.473
23.419
13.271
20.487
42.673
33.507
32.776
45.363
69.522
115.529
X2Y
6.51
12.76
13.68
14.69
15.54
16.65
18.40
20.91
22.14
20.14
23.40
31.71
32.12
31.28
38.16
47.00
59.80
X1X2
101.85
244.42
228.72
239.59
229.74
233.85
275.68
352.07
342.72
237.88
350.20
593.46
504.90
554.30
684.72
924.50
1305.98
22.5200
384.320
303.000
31.7058
10083.0
5865.00
559.216
424.890
7404.58
1.32471
22.6071
17.8235
1394.69
464.471
1.87342
50.1046
23.5041
554.641
17.0000
Regression Analysis: GP6_1 versus CP6_1
Note that p-values are very low implying that
coefficients are significant.
The regression equation is
GP6_1 = 0.513 + 0.0359 CP6_1
Predictor
Coef
SE Coef
Constant
0.51254
0.04562
CP6_1
0.035925 0.001873
S = 0.0699550
R-Sq = 96.1%
T
P
11.24 0.000
19.18 0.000
R-Sq(adj) = 95.8%
34
252y0771t 11/28/07
Analysis of Variance
A low p-value means to reject hypothesis that there is no relation
between dependent variables and independent variables.
Source
DF
SS
MS
F
P
Regression
1 1.8000 1.8000 367.82 0.000
Residual Error 15 0.0734 0.0049
Total
16 1.8734
Unusual Observations
Obs CP6_1
GP6_1
Fit SE Fit Residual St Resid
2
22.2 1.1600 1.3108 0.0170
-0.1508
-2.22R
17
50.2 2.3000 2.3171 0.0545
-0.0171
-0.39 X
R denotes an observation with a large standardized residual.
X denotes an observation whose X value gives it large influence.
Regression Analysis: GP6_1 versus Yr6_1
Note that p-values are below 5% implying that
coefficients are significant at 5% level.
The regression equation is
GP6_1 = 0.423 + 0.0506 Yr6_1
Predictor
Coef
SE Coef
T
P
Constant
0.4228
0.1840 2.30 0.036
Yr6_1
0.050604 0.009909 5.11 0.000
S = 0.213544
R-Sq = 63.5%
R-Sq(adj) = 61.1%
Analysis of Variance
A low p-value means to reject hypothesis that there is no relation
between dependent variables and independent variables.
Source
DF
SS
MS
F
P
Regression
1 1.1894 1.1894 26.08 0.000
Residual Error 15 0.6840 0.0456
Total
16 1.8734
Unusual Observations
Obs Yr6_1
GP6_1
Fit SE Fit Residual St Resid
17
26.0 2.3000 1.7385 0.0962
0.5615
2.95R
R denotes an observation with a large standardized residual.
Regression Analysis: GP6_1 versus CP6_1, Yr6_1
Note that p-values are very low implying
that coefficients are significant.
The regression equation is
GP6_1 = 0.383 + 0.0301 CP6_1 + 0.0147 Yr6_1
Predictor
Coef
SE Coef
Constant
0.38294
0.03335
CP6_1
0.030090 0.001428
Yr6_1
0.014672 0.002474
S = 0.0386385
R-Sq = 98.9%
T
P
11.48 0.000
21.08 0.000
5.93 0.000
R-Sq(adj) = 98.7%
Analysis of Variance
A low p-value means to reject hypothesis that there is no relation
between dependent variables and independent variables.
Source
DF
SS
MS
F
P
Regression
2 1.85252 0.92626 620.43 0.000
Residual Error 14 0.02090 0.00149
Total
16 1.87342
Source
CP6_1
Yr6_1
DF
1
1
Seq SS
1.80002
0.05250
Unusual Observations
Obs CP6_1
GP6_1
Fit
SE Fit Residual St Resid
14
24.1 1.36000 1.44558 0.01479 -0.08558
-2.40R
17
50.2 2.30000 2.27586 0.03087
0.02414
1.04 X
R denotes an observation with a large standardized residual.
X denotes an observation whose X value gives it large influence.
35
252y0771t 11/28/07
MTB > WOpen "C:\Documents and Settings\rbove\My Documents\Minitab\252x077117a.MTW".
Retrieving worksheet from file: 'C:\Documents and Settings\rbove\My
Documents\Minitab\252x0771-17a.MTW'
Worksheet was saved on Wed Nov 28 2007
Version 7
Results for: 252x0771-17a.MTW
MTB > Execute "C:\Documents and Settings\RBOVE\My
Documents\Minitab\SpareParts.mtb" 1.
Executing from file: C:\Documents and Settings\RBOVE\My
Documents\Minitab\SpareParts.mtb
Data Display
Row
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
GP7_1
0.95
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
CP7_1
17.90
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
Yr7_1
8
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
Ysq
0.9025
1.3456
1.2996
1.2769
1.2321
1.2321
1.3225
1.5129
1.5129
1.1236
1.3689
2.2801
2.1316
1.8496
2.5281
3.5344
5.2900
X1sq
320.41
493.73
363.28
339.66
269.29
243.05
296.87
428.90
362.52
156.75
306.60
798.63
526.70
580.81
813.96
1367.52
2523.05
Data Display
Sums of columns above.
Data Display
Spare Parts computation
sumY
sumX1
sumX2
sumYsq
sumX1sq
sumX2sq
sumX1y
sumX2y
sumx1x2
meanY
meanX1
meanX2
SSX1
SSX2
SSY
SX1Y
SX2Y
SX1X2
n
X2sq
64
121
144
169
196
225
256
289
324
361
400
441
484
529
576
625
676
X1Y
17.005
25.775
21.728
20.826
18.215
17.305
19.815
25.473
23.419
13.271
20.487
42.673
33.507
32.776
45.363
69.522
115.529
X2Y
7.60
12.76
13.68
14.69
15.54
16.65
18.40
20.91
22.14
20.14
23.40
31.71
32.12
31.28
38.16
47.00
59.80
X1X2
143.20
244.42
228.72
239.59
229.74
233.85
275.68
352.07
342.72
237.88
350.20
593.46
504.90
554.30
684.72
924.50
1305.98
22.5400
387.670
304.000
31.7434
10191.7
5880.00
562.689
425.980
7445.93
1.32588
22.8041
17.8824
1351.27
443.765
1.85801
48.6843
22.9118
513.478
17.0000
Regression Analysis: GP7_1 versus CP7_1 Note that p-values are very low implying that
coefficients are significant.
The regression equation is
GP7_1 = 0.504 + 0.0360 CP7_1
Predictor
Coef
SE Coef
Constant
0.50429
0.05546
CP7_1
0.036028 0.002265
S = 0.0832640
R-Sq = 94.4%
T
P
9.09 0.000
15.91 0.000
R-Sq(adj) = 94.0%
36
252y0771t 11/28/07
Analysis of Variance
A low p-value means to reject hypothesis that there is no relation
between dependent variables and independent variables.
Source
DF
SS
MS
F
P
Regression
1 1.7540 1.7540 253.00 0.000
Residual Error 15 0.1040 0.0069
Total
16 1.8580
Unusual Observations
Obs CP7_1
GP7_1
Fit SE Fit Residual St Resid
1
17.9 0.9500 1.1492 0.0230
-0.1992
-2.49R
17
50.2 2.3000 2.3140 0.0653
-0.0140
-0.27 X
R denotes an observation with a large standardized residual.
X denotes an observation whose X value gives it large influence.
Regression Analysis: GP7_1 versus Yr7_1
Note that p-values are below 5% implying that
coefficients are significant at 5% level.
The regression equation is
GP7_1 = 0.403 + 0.0516 Yr7_1
Predictor
Coef SE Coef
Constant
0.4026
0.1873
Yr7_1
0.05163 0.01007
S = 0.212143
R-Sq = 63.7%
T
P
2.15 0.048
5.13 0.000
R-Sq(adj) = 61.2%
Analysis of Variance
A low p-value means to reject hypothesis that there is no relation between
dependent variables and independent variables.
Source
DF
SS
MS
F
P
Regression
1 1.1829 1.1829 26.29 0.000
Residual Error 15 0.6751 0.0450
Total
16 1.8580
Unusual Observations
Obs Yr7_1
GP7_1
Fit SE Fit Residual St Resid
17
26.0 2.3000 1.7450 0.0966
0.5550
2.94R
R denotes an observation with a large standardized residual.
Regression Analysis: GP7_1 versus CP7_1, Yr7_1
Note that p-values are very low
implying that coefficients are significant.
The regression equation is
GP7_1 = 0.341 + 0.0293 CP7_1 + 0.0177 Yr7_1
Predictor
Coef
SE Coef
Constant
0.34074
0.03797
CP7_1
0.029286 0.001557
Yr7_1
0.017744 0.002718
S = 0.0428518
R-Sq = 98.6%
T
P
8.97 0.000
18.81 0.000
6.53 0.000
R-Sq(adj) = 98.4%
Analysis of Variance
A low p-value means to reject hypothesis that there is no relation between
dependent variables and independent variables.
Source
DF
SS
MS
F
P
Regression
2 1.83230 0.91615 498.92 0.000
Residual Error 14 0.02571 0.00184
Total
16 1.85801
Source
CP7_1
Yr7_1
DF
1
1
Seq SS
1.75402
0.07829
Unusual Observations
Obs CP7_1
GP7_1
Fit SE Fit Residual St Resid
14
24.1 1.3600 1.4546 0.0164
-0.0946
-2.39R
17
50.2 2.3000 2.2731 0.0342
0.0269
1.04 X
R denotes an observation with a large standardized residual.
X denotes an observation whose X value gives it large influence.
37
252y0771t 11/28/07
MTB > WOpen "C:\Documents and Settings\rbove\My Documents\Minitab\252x077118a.MTW".
Retrieving worksheet from file: 'C:\Documents and Settings\rbove\My
Documents\Minitab\252x0771-18a.MTW'
Worksheet was saved on Wed Nov 28 2007
Version 8
Results for: 252x0771-18a.MTW
MTB > Execute "C:\Documents and Settings\RBOVE\My
Documents\Minitab\SpareParts.mtb" 1.
Executing from file: C:\Documents and Settings\RBOVE\My
Documents\Minitab\SpareParts.mtb
Data Display
Row
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
GP8_1
0.96
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
CP8_1
14.67
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
Yr8_1
9
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
Ysq
0.9216
1.3456
1.2996
1.2769
1.2321
1.2321
1.3225
1.5129
1.5129
1.1236
1.3689
2.2801
2.1316
1.8496
2.5281
3.5344
5.2900
X1sq
215.21
493.73
363.28
339.66
269.29
243.05
296.87
428.90
362.52
156.75
306.60
798.63
526.70
580.81
813.96
1367.52
2523.05
Data Display
Sums of columns above.
Data Display
Spare Parts computation
sumY
sumX1
sumX2
sumYsq
sumX1sq
sumX2sq
sumX1y
sumX2y
sumx1x2
meanY
meanX1
meanX2
SSX1
SSX2
SSY
SX1Y
SX2Y
SX1X2
n
X2sq
81
121
144
169
196
225
256
289
324
361
400
441
484
529
576
625
676
X1Y
14.083
25.775
21.728
20.826
18.215
17.305
19.815
25.473
23.419
13.271
20.487
42.673
33.507
32.776
45.363
69.522
115.529
X2Y
8.64
12.76
13.68
14.69
15.54
16.65
18.40
20.91
22.14
20.14
23.40
31.71
32.12
31.28
38.16
47.00
59.80
X1X2
132.03
244.42
228.72
239.59
229.74
233.85
275.68
352.07
342.72
237.88
350.20
593.46
504.90
554.30
684.72
924.50
1305.98
22.5500
384.440
305.000
31.7625
10086.5
5897.00
559.767
427.020
7434.76
1.32647
22.6141
17.9412
1392.77
424.941
1.85059
49.8189
22.4465
537.454
17.0000
Regression Analysis: GP8_1 versus CP8_1
Note that p-values are very low implying that
coefficients are significant.
The regression equation is
GP8_1 = 0.518 + 0.0358 CP8_1
Predictor
Coef
SE Coef
Constant
0.51757
0.04413
CP8_1
0.035770 0.001812
S = 0.0676191
R-Sq = 96.3%
T
P
11.73 0.000
19.74 0.000
R-Sq(adj) = 96.0%
38
252y0771t 11/28/07
Analysis of Variance
A low p-value means to reject hypothesis that there is no relation between
dependent variables and independent variables.
Source
DF
SS
MS
F
P
Regression
1 1.7820 1.7820 389.73 0.000
Residual Error 15 0.0686 0.0046
Total
16 1.8506
Unusual Observations
Obs CP8_1
GP8_1
Fit SE Fit Residual St Resid
2
22.2 1.1600 1.3124 0.0164
-0.1524
-2.32R
17
50.2 2.3000 2.3143 0.0527
-0.0143
-0.34 X
R denotes an observation with a large standardized residual.
X denotes an observation whose X value gives it large influence.
Regression Analysis: GP8_1 versus Yr8_1
Note that p-value for slope is very low
implying that coefficient is significant. The constant is not significant at the 5% level.
The regression equation is
GP8_1 = 0.379 + 0.0528 Yr8_1
Predictor
Coef SE Coef
Constant
0.3788
0.1902
Yr8_1
0.05282 0.01021
S = 0.210540
R-Sq = 64.1%
T
P
1.99 0.065
5.17 0.000
R-Sq(adj) = 61.7%
Analysis of Variance
A low p-value means to reject hypothesis that there is no relation
between dependent variables and independent variables.
Source
DF
SS
MS
F
P
Regression
1 1.1857 1.1857 26.75 0.000
Residual Error 15 0.6649 0.0443
Total
16 1.8506
Unusual Observations
Obs Yr8_1
GP8_1
Fit SE Fit Residual St Resid
17
26.0 2.3000 1.7522 0.0969
0.5478
2.93R
R denotes an observation with a large standardized residual.
Regression Analysis: GP8_1 versus CP8_1, Yr8_1 Note that p-values are very low implying
that coefficients are significant.
The regression equation is
GP8_1 = 0.381 + 0.0301 CP8_1 + 0.0148 Yr8_1
Predictor
Coef
SE Coef
Constant
0.38110
0.03488
CP8_1
0.030054 0.001446
Yr8_1
0.014811 0.002617
S = 0.0386061
R-Sq = 98.9%
T
P
10.93 0.000
20.79 0.000
5.66 0.000
R-Sq(adj) = 98.7%
Analysis of Variance
A low p-value means to reject hypothesis that there is no relation between
dependent variables and independent variables.
Source
DF
SS
MS
F
P
Regression
2 1.82972 0.91486 613.82 0.000
Residual Error 14 0.02087 0.00149
Total
16 1.85059
Source
CP8_1
Yr8_1
DF
1
1
Seq SS
1.78200
0.04772
Unusual Observations
Obs CP8_1
GP8_1
Fit
SE Fit Residual St Resid
14
24.1 1.36000 1.44605 0.01510 -0.08605
-2.42R
17
50.2 2.30000 2.27580 0.03082
0.02420
1.04 X
R denotes an observation with a large standardized residual.
X denotes an observation whose X value gives it large influence.
39
252y0771t 11/28/07
MTB > WOpen "C:\Documents and Settings\rbove\My Documents\Minitab\252x077119a.MTW".
Retrieving worksheet from file: 'C:\Documents and Settings\rbove\My
Documents\Minitab\252x0771-19a.MTW'
Worksheet was saved on Tue Nov 27 2007
Version 9
Results for: 252x0771-19a.MTW
MTB > Execute "C:\Documents and Settings\RBOVE\My
Documents\Minitab\SpareParts.mtb" 1.
Executing from file: C:\Documents and Settings\RBOVE\My
Documents\Minitab\SpareParts.mtb
Data Display
Row
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
GP9
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
CP9
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
Yr9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
ysq
1.0404
1.3456
1.2996
1.2769
1.2321
1.2321
1.3225
1.5129
1.5129
1.1236
1.3689
2.2801
2.1316
1.8496
2.5281
3.5344
5.2900
x1sq
322.92
493.73
363.28
339.66
269.29
243.05
296.87
428.90
362.52
156.75
306.60
798.63
526.70
580.81
813.96
1367.52
2523.05
x2sq
100
121
144
169
196
225
256
289
324
361
400
441
484
529
576
625
676
Data Display
Sums of columns above.
Data Display
Spare Parts computation
sumY
sumX1
sumX2
sumYsq
sumX1sq
sumX2sq
sumX1y
sumX2y
sumx1x2
meanY
meanX1
meanX2
SSx1
SSx2
SSY
SX1Y
SX2Y
SX1X2
n
x1y
18.329
25.775
21.728
20.826
18.215
17.305
19.815
25.473
23.419
13.271
20.487
42.673
33.507
32.776
45.363
69.522
115.529
x2y
10.20
12.76
13.68
14.69
15.54
16.65
18.40
20.91
22.14
20.14
23.40
31.71
32.12
31.28
38.16
47.00
59.80
x1x2
179.70
244.42
228.72
239.59
229.74
233.85
275.68
352.07
342.72
237.88
350.20
593.46
504.90
554.30
684.72
924.50
1305.98
22.6100
387.740
306.000
31.8813
10194.3
5916.00
564.014
428.580
7482.43
1.33000
22.8082
18.0000
1350.59
408.000
1.81000
48.3193
21.6000
503.110
17.0000
Regression Analysis: GP9 versus CP9
Note that p-values are very low implying that coefficients
are significant.
The regression equation is
GP9 = 0.514 + 0.0358 CP9
Predictor
Coef
SE Coef
Constant
0.51400
0.04906
CP9
0.035776 0.002003
S = 0.0736254
R-Sq = 95.5%
T
P
10.48 0.000
17.86 0.000
R-Sq(adj) = 95.2%
40
252y0771t 11/28/07
Analysis of Variance
A low p-value means to reject hypothesis that there is no relation between
dependent variables and independent variables.
Source
DF
SS
MS
F
P
Regression
1 1.7287 1.7287 318.91 0.000
Residual Error 15 0.0813 0.0054
Total
16 1.8100
Unusual Observations
Obs
CP9
GP9
Fit SE Fit Residual St Resid
2 22.2 1.1600 1.3090 0.0179
-0.1490
-2.09R
17 50.2 2.3000 2.3111 0.0578
-0.0111
-0.24 X
R denotes an observation with a large standardized residual.
X denotes an observation whose X value gives it large influence.
Regression Analysis: GP9 versus Yr9
Note that p-value for slope is very low implying that
coefficient is significant. The constant is not significant at the 5% level.
The regression equation is
GP9 = 0.377 + 0.0529 Yr9
Predictor
Coef SE Coef
Constant
0.3771
0.1947
Yr9
0.05294 0.01044
S = 0.210788
R-Sq = 63.2%
T
P
1.94 0.072
5.07 0.000
R-Sq(adj) = 60.7%
Analysis of Variance
A low p-value means to reject hypothesis that there is no relation
between dependent variables and independent variables.
Source
DF
SS
MS
F
P
Regression
1 1.1435 1.1435 25.74 0.000
Residual Error 15 0.6665 0.0444
Total
16 1.8100
Unusual Observations
Obs
Yr9
GP9
Fit SE Fit Residual St Resid
17 26.0 2.3000 1.7535 0.0979
0.5465
2.93R
R denotes an observation with a large standardized residual.
Regression Analysis: GP9 versus CP9, Yr9
Note that p-values are very low implying that
coefficients are significant.
The regression equation is
GP9 = 0.359 + 0.0297 CP9 + 0.0163 Yr9
Predictor
Coef
SE Coef
Constant
0.35888
0.03707
CP9
0.029696 0.001485
Yr9
0.016323 0.002702
S = 0.0401248
R-Sq = 98.8%
T
P
9.68 0.000
20.00 0.000
6.04 0.000
R-Sq(adj) = 98.6%
Analysis of Variance
A low p-value means to reject hypothesis that there is no relation between
dependent variables and independent variables.
Source
DF
SS
MS
F
P
Regression
2 1.78746 0.89373 555.11 0.000
Residual Error 14 0.02254 0.00161
Total
16 1.81000
Source
CP9
Yr9
DF
1
1
Seq SS
1.72869
0.05877
Unusual Observations
Obs
CP9
GP9
Fit
SE Fit Residual St Resid
14 24.1 1.36000 1.44997 0.01568 -0.08997
-2.44R
17 50.2 2.30000 2.27490 0.03205
0.02510
1.04 X
R denotes an observation with a large standardized residual.
X denotes an observation whose X value gives it large influence.
41
252y0771t 11/28/07
Appendix B ANOVA Problem
————— 11/30/2007 11:27:56 PM ————————————————————
Welcome to Minitab, press F1 for help.
Version 0
MTB > WOpen "C:\Documents and Settings\RBOVE\My Documents\Minitab\252x07712a1.MTW".
Retrieving worksheet from file: 'C:\Documents and Settings\RBOVE\My
Documents\Minitab\252x0771-2a1.MTW'
Worksheet was saved on Thu Nov 29 2007
MTB > Execute "C:\Documents and Settings\RBOVE\My
Documents\Minitab\ANOVA252y077.mtb" 1.
Executing from file: C:\Documents and Settings\RBOVE\My
Documents\Minitab\ANOVA252y077.mtb
Data Display
Row
1
2
3
4
Fifi
36
22
19
16
Adelina
39
14
20
18
Maria
21
32
28
22
Data Display Display for ANOVA tableau. Includes Total sum, Total sum of squares, grand mean, SST, SSB =
SSC, Sums of squared row and column means.
SumAll
287.000
SumAllsq
7571.00
grandM
23.9167
SST
706.917
SSC
20.6667
SSR
290.917
SColMsq
1721.19
SRowMsq
2385.00
* NOTE * One or more variables are undefined.
Data Display Display for ANOVA tableau. Includes Row sums, Column sums, Row and Column sum of squares,
Row and column means and row and column means squared.
Row
1
2
3
4
RowS
96
68
67
56
ColS
93
91
103
RowSS
3258
1704
1545
1064
ColSS
2397
2441
2733
RowMean
32.0000
22.6667
22.3333
18.6667
ColMean
23.25
22.75
25.75
ColMsq
540.563
517.563
663.063
RowMsq
1024.00
513.78
498.78
348.44
One-way ANOVA: Fifi, Adelina, Maria Test of equality of column means. High p-value means do not
reject equality.
Source DF
Factor
2
Error
9
Total
11
S = 8.732
Level
Fifi
Adelina
Maria
N
4
4
4
SS
MS
F
P
20.7 10.3 0.14 0.875
686.3 76.3
706.9
R-Sq = 2.92%
R-Sq(adj) = 0.00%
Mean
23.250
22.750
25.750
StDev
8.846
11.117
5.188
Individual 95% CIs For Mean Based on
Pooled StDev
---------+---------+---------+---------+
(----------------*---------------)
(----------------*---------------)
(----------------*---------------)
---------+---------+---------+---------+
18.0
24.0
30.0
36.0
Pooled StDev = 8.732
42
252y0771t 11/28/07
Two-way ANOVA: CCalls versus Venue, Soprano Test of equality of row and column means.
High p-value means do not reject equality.
Source
DF
SS
MS
F
P
Venue
3 290.917 96.9722 1.47 0.314
Soprano
2
20.667 10.3333 0.16 0.858
Error
6 395.333 65.8889
Total
11 706.917
S = 8.117
R-Sq = 44.08%
R-Sq(adj) = 0.00%
Individual 95% CIs For Mean Based on
Pooled StDev
Venue
Mean ---+---------+---------+---------+-----Milan
32.0000
(----------*----------)
Naples 22.3333
(----------*-----------)
Rome
18.6667 (-----------*----------)
Venice 22.6667
(-----------*----------)
---+---------+---------+---------+-----10
20
30
40
Individual 95% CIs For Mean Based on
Pooled StDev
Soprano
Mean ---------+---------+---------+---------+
Adelina 22.75 (----------------*---------------)
Fifi
23.25
(----------------*---------------)
Maria
25.75
(----------------*---------------)
---------+---------+---------+---------+
18.0
24.0
30.0
36.0
Data Display Display for Kruskal-Wallis test. Alternate columns give ranks.
Row Fifi rCCalls_Fifi Adelina rCCalls_Adelina Maria rCCalls_Maria
1
36
11.0
39
12
21
6.0
2
22
7.5
14
1
32
10.0
3
19
4.0
20
5
28
9.0
4
16
2.0
18
3
22
7.5
Data Display Display for Kruskal-Wallis test. Gives sums of column ranks.
rs1
rs2
rs3
24.5000
21.0000
32.5000
Kruskal-Wallis Test: CCalls versus Soprano Test of equality of medians. High p-value means do
not reject equality of medians.
Kruskal-Wallis Test on CCalls
Ave
Soprano
N Median Rank
Z
Adelina
4
19.00
5.3 -0.85
Fifi
4
20.50
6.1 -0.25
Maria
4
25.00
8.1
1.10
Overall 12
6.5
H = 1.34 DF = 2 P = 0.513
H = 1.34 DF = 2 P = 0.511 (adjusted for ties)
* NOTE * One or more small samples
Data Display Display for Friedman test. Alternate columns give ranks within rows.
Row
1
2
3
4
Fifi
36
22
19
16
C31
2
2
1
1
Adelina
39
14
20
18
C32
3
1
2
2
Maria
21
32
28
22
C33
1
3
3
3
Data Display Display for Friedman test. Gives sums of ranks by column.
rs1
rs2
rs3
6.00000
8.00000
10.0000
43
252y0771t 11/28/07
Friedman Test: CCalls versus Soprano blocked by Venue Test of equality of medians blocked
by Venue. High p-value means do not reject equality of medians.
S = 2.00 DF = 2 P = 0.368
Sum of
Soprano N Est Median
Ranks
Adelina 4
22.000
8.0
Fifi
4
20.500
6.0
Maria
4
28.000
10.0
Grand median = 23.500
Data Display Display for Levene test. Original data.
Row
1
2
3
4
Fifi_1
36
22
19
16
Adelina_1
39
14
20
18
Maria_1
21
32
28
22
Data Display Display for Levene test. Column medians.
K41
K42
K43
20.5000
19.0000
25.0000
Data Display Display for Levene test. Columns less medians.
Row
1
2
3
4
Fifi_1
15.5
1.5
-1.5
-4.5
Adelina_1
20
-5
1
-1
Maria_1
-4
7
3
-3
Data Display Display for Levene test. Absolute value of Columns less medians.
Row
1
2
3
4
Fifi_1
15.5
1.5
1.5
4.5
Adelina_1
20
5
1
1
Maria_1
4
7
3
3
One-way ANOVA: Fifi_1, Adelina_1, Maria_1 Test of equality of variances. High p-value means do
not reject equality of variances.
Source DF
Factor
2
Error
9
Total
11
S = 6.568
Level
Fifi_1
Adelina_1
Maria_1
SS
MS
F
P
12.7
6.3 0.15 0.865
388.3 43.1
400.9
R-Sq = 3.16%
R-Sq(adj) = 0.00%
N
4
4
4
Mean
5.750
6.750
4.250
StDev
6.652
9.032
1.893
Individual 90% CIs For Mean Based on
Pooled StDev
----+---------+---------+---------+----(--------------*--------------)
(--------------*--------------)
(--------------*--------------)
----+---------+---------+---------+----0.0
4.0
8.0
12.0
Pooled StDev = 6.568
44
252y0771t 11/28/07
Test for Equal Variances: CCalls versus Soprano Test of equality of variances. High p-value
means do not reject equality of variances.
90% Bonferroni confidence intervals for standard deviations
Soprano N
Lower
StDev
Upper
Adelina 4 6.01846 11.1168 47.6964
Fifi 4 4.78903
8.8459 37.9532
Maria 4 2.80877
5.1881 22.2596
Bartlett's Test (Normal Distribution)
Test statistic = 1.39, p-value = 0.499
Levene's Test (Any Continuous Distribution)
Test statistic = 0.15, p-value = 0.865
Test for Equal Variances: CCalls versus Soprano
45
252y0771t 11/28/07
Version 1
MTB > WOpen "C:\Documents and Settings\RBOVE\My Documents\Minitab\252x07712a2.MTW".
Retrieving worksheet from file: 'C:\Documents and Settings\RBOVE\My
Documents\Minitab\252x0771-2a2.MTW'
Worksheet was saved on Thu Nov 29 2007
Results for: 252x0771-2a2.MTW
MTB > Execute "C:\Documents and Settings\RBOVE\My
Documents\Minitab\ANOVA252y077.mtb" 1.
Executing from file: C:\Documents and Settings\RBOVE\My
Documents\Minitab\ANOVA252y077.mtb
Data Display
Row
1
2
3
4
Fifi
37
22
19
16
Adelina
40
14
20
18
Maria
22
32
28
22
Data Display Display for ANOVA tableau. Includes Total sum, Total sum of squares, grand mean, SST, SSB =
SSC, Sums of squared row and column means.
SumAll
290.000
SumAllsq
7766.00
grandM
24.1667
SST
757.667
SSC
20.6667
SSR
341.667
SColMsq
1757.25
SRowMsq
2450.00
* NOTE * One or more variables are undefined.
Data Display Display for ANOVA tableau. Includes Row sums, Column sums, Row and Column sum of
squares, Row and column means and row and column means squared.
Row RowS ColS RowSS ColSS RowMean ColMean
1
99
94
3453
2470 33.0000
23.5
2
68
92
1704
2520 22.6667
23.0
3
67
104
1545
2776 22.3333
26.0
4
56
1064
18.6667
ColMsq
552.25
529.00
676.00
RowMsq
1089.00
513.78
498.78
348.44
One-way ANOVA: Fifi, Adelina, Maria Test of equality of column means. High p-value means do not
reject equality.
Source DF
Factor
2
Error
9
Total
11
S = 9.049
Level
Fifi
Adelina
Maria
N
4
4
4
SS
MS
F
P
20.7 10.3 0.13 0.883
737.0 81.9
757.7
R-Sq = 2.73%
R-Sq(adj) = 0.00%
Mean
23.500
23.000
26.000
StDev
9.327
11.605
4.899
Individual 95% CIs For Mean Based on
Pooled StDev
---------+---------+---------+---------+
(----------------*----------------)
(----------------*----------------)
(----------------*----------------)
---------+---------+---------+---------+
18.0
24.0
30.0
36.0
Pooled StDev = 9.049
46
252y0771t 11/28/07
Two-way ANOVA: CCalls versus Venue, Soprano Test of equality of row and column means.
High p-value means do not reject equality.
Source
DF
SS
MS
F
P
Venue
3 341.667 113.889 1.73 0.260
Soprano
2
20.667
10.333 0.16 0.858
Error
6 395.333
65.889
Total
11 757.667
S = 8.117
R-Sq = 47.82%
R-Sq(adj) = 4.34%
Venue
Milan
Naples
Rome
Venice
Mean
33.0000
22.3333
18.6667
22.6667
Individual 95% CIs For Mean Based on
Pooled StDev
---+---------+---------+---------+-----(----------*----------)
(----------*-----------)
(-----------*----------)
(-----------*----------)
---+---------+---------+---------+-----10
20
30
40
Individual 95% CIs For Mean Based on
Pooled StDev
Soprano Mean --------+---------+---------+---------+Adelina 23.0 (---------------*----------------)
Fifi
23.5
(---------------*----------------)
Maria
26.0
(---------------*----------------)
--------+---------+---------+---------+18.0
24.0
30.0
36.0
Data Display Display for Kruskal-Wallis test. Alternate columns give ranks.
Row Fifi rCCalls_Fifi Adelina rCCalls_Adelina Maria rCCalls_Maria
1
37
11
40
12
22
7
2
22
7
14
1
32
10
3
19
4
20
5
28
9
4
16
2
18
3
22
7
Data Display Display for Kruskal-Wallis test. Gives sums of column ranks.
rs1
rs2
rs3
24.0000
21.0000
33.0000
Kruskal-Wallis Test: CCalls versus Soprano Test of equality of medians. High p-value means do
not reject equality of medians.
Kruskal-Wallis Test on CCalls
Ave
Soprano
N Median Rank
Z
Adelina
4
19.00
5.3 -0.85
Fifi
4
20.50
6.0 -0.34
Maria
4
25.00
8.3
1.19
Overall 12
6.5
H = 1.50 DF = 2 P = 0.472
H = 1.52 DF = 2 P = 0.467 (adjusted for ties)
* NOTE * One or more small samples
Data Display Display for Friedman test. Alternate columns give ranks within rows.
Row
1
2
3
4
Fifi
37
22
19
16
C31
2
2
1
1
Adelina
40
14
20
18
C32
3
1
2
2
Maria
22
32
28
22
C33
1
3
3
3
Data Display Display for Friedman test. Gives sums of ranks by column.
rs1
rs2
rs3
6.00000
8.00000
10.0000
47
252y0771t 11/28/07
Friedman Test: CCalls versus Soprano blocked by Venue Test of equality of medians
blocked by Venue. High p-value means do not reject equality of medians.
S = 2.00 DF = 2 P = 0.368
Sum of
Soprano N Est Median
Ranks
Adelina 4
22.000
8.0
Fifi
4
20.500
6.0
Maria
4
28.000
10.0
Grand median = 23.500
Data Display Display for Levene test. Original data.
Row
1
2
3
4
Fifi_1
37
22
19
16
Adelina_1
40
14
20
18
Maria_1
22
32
28
22
Data Display Display for Levene test. Column medians.
K41
K42
K43
20.5000
19.0000
25.0000
Data Display Display for Levene test. Columns less medians.
Row
1
2
3
4
Fifi_1
16.5
1.5
-1.5
-4.5
Adelina_1
21
-5
1
-1
Maria_1
-3
7
3
-3
Data Display Display for Levene test. Absolute value of Columns less medians.
Row
1
2
3
4
Fifi_1
16.5
1.5
1.5
4.5
Adelina_1
21
5
1
1
Maria_1
3
7
3
3
One-way ANOVA: Fifi_1, Adelina_1, Maria_1 Test of equality of variances. High p-value means do
not reject equality of variances.
Source DF
SS
MS
F
P
Factor
2
18.7
9.3 0.19 0.828
Error
9 437.0 48.6
Total
11 455.7
S = 6.968
R-Sq = 4.10%
R-Sq(adj) = 0.00%
Individual 90% CIs For Mean Based on
Pooled StDev
Level
N
Mean StDev ------+---------+---------+---------+--Fifi_1
4 6.000 7.141
(---------------*---------------)
Adelina_1 4 7.000 9.522
(--------------*---------------)
Maria_1
4 4.000 2.000 (---------------*---------------)
------+---------+---------+---------+--0.0
4.0
8.0
12.0
Pooled StDev = 6.968
48
252y0771t 11/28/07
Test for Equal Variances: CCalls versus Soprano Test of equality of variances. High p-value
means do not reject equality of variances.
90% Bonferroni confidence intervals for standard deviations
Soprano N
Lower
StDev
Upper
Adelina 4 6.28255 11.6046 49.7893
Fifi 4 5.04970
9.3274 40.0189
Maria 4 2.65223
4.8990 21.0190
Bartlett's Test (Normal Distribution)
Test statistic = 1.75, p-value = 0.417
Levene's Test (Any Continuous Distribution)
Test statistic = 0.19, p-value = 0.828
Test for Equal Variances: CCalls versus Soprano
49
252y0771t 11/28/07
Version 2
MTB > WOpen "C:\Documents and Settings\RBOVE\My Documents\Minitab\252x07712a3.MTW".
Retrieving worksheet from file: 'C:\Documents and Settings\RBOVE\My
Documents\Minitab\252x0771-2a3.MTW'
Worksheet was saved on Thu Nov 29 2007
Results for: 252x0771-2a3.MTW
MTB > Execute "C:\Documents and Settings\RBOVE\My
Documents\Minitab\ANOVA252y077.mtb" 1.
Executing from file: C:\Documents and Settings\RBOVE\My
Documents\Minitab\ANOVA252y077.mtb
Data Display
Row
1
2
3
4
Fifi
38
22
19
16
Adelina
41
14
20
18
Maria
23
32
28
22
Data Display Display for ANOVA tableau. Includes Total sum, Total sum of squares, grand mean, SST, SSB =
SSC, Sums of squared row and column means.
SumAll
293.000
SumAllsq
7967.00
grandM
24.4167
SST
812.917
SSC
20.6667
SSR
396.917
SColMsq
1793.69
SRowMsq
2517.00
* NOTE * One or more variables are undefined.
Data Display Display for ANOVA tableau. Includes Row sums, Column sums, Row and Column sum of
squares, Row and column means and row and column means squared.
Row RowS ColS RowSS ColSS RowMean ColMean
1
102
95
3654
2545 34.0000
23.75
2
68
93
1704
2601 22.6667
23.25
3
67
105
1545
2821 22.3333
26.25
4
56
1064
18.6667
ColMsq
564.063
540.563
689.063
RowMsq
1156.00
513.78
498.78
348.44
One-way ANOVA: Fifi, Adelina, Maria Test of equality of column means. High p-value means do not
reject equality.
Source DF
Factor
2
Error
9
Total
11
S = 9.382
Level
Fifi
Adelina
Maria
N
4
4
4
SS
MS
F
P
20.7 10.3 0.12 0.891
792.3 88.0
812.9
R-Sq = 2.54%
R-Sq(adj) = 0.00%
Mean
23.750
23.250
26.250
StDev
9.811
12.093
4.646
Individual 95% CIs For Mean Based on
Pooled StDev
--+---------+---------+---------+------(--------------*--------------)
(--------------*--------------)
(---------------*--------------)
--+---------+---------+---------+------14.0
21.0
28.0
35.0
Pooled StDev = 9.382
50
252y0771t 11/28/07
Two-way ANOVA: CCalls versus Venue, Soprano Test of equality of row and column means.
High p-value means do not reject equality.
Source
DF
SS
MS
F
P
Venue
3 396.917 132.306 2.01 0.214
Soprano
2
20.667
10.333 0.16 0.858
Error
6 395.333
65.889
Total
11 812.917
S = 8.117
R-Sq = 51.37%
R-Sq(adj) = 10.84%
Venue
Milan
Naples
Rome
Venice
Mean
34.0000
22.3333
18.6667
22.6667
Individual 95% CIs For Mean Based on
Pooled StDev
---+---------+---------+---------+-----(----------*----------)
(----------*-----------)
(-----------*----------)
(-----------*----------)
---+---------+---------+---------+-----10
20
30
40
Individual 95% CIs For Mean Based on
Pooled StDev
Soprano
Mean --------+---------+---------+---------+Adelina 23.25 (----------------*---------------)
Fifi
23.75
(----------------*---------------)
Maria
26.25
(----------------*---------------)
--------+---------+---------+---------+18.0
24.0
30.0
36.0
Data Display Display for Kruskal-Wallis test. Alternate columns give ranks.
Row Fifi rCCalls_Fifi Adelina rCCalls_Adelina Maria rCCalls_Maria
1
38
11.0
41
12
23
8.0
2
22
6.5
14
1
32
10.0
3
19
4.0
20
5
28
9.0
4
16
2.0
18
3
22
6.5
Data Display Display for Kruskal-Wallis test. Gives sums of column ranks.
rs1
rs2
rs3
23.5000
21.0000
33.5000
Kruskal-Wallis Test: CCalls versus Soprano Test of equality of medians. High p-value means do
not reject equality of medians.
Kruskal-Wallis Test on CCalls
Ave
Soprano
N Median Rank
Z
Adelina
4
19.00
5.3 -0.85
Fifi
4
20.50
5.9 -0.42
Maria
4
25.50
8.4
1.27
Overall 12
6.5
H = 1.68 DF = 2 P = 0.431
H = 1.69 DF = 2 P = 0.430 (adjusted for ties)
* NOTE * One or more small samples
Data Display Display for Friedman test. Alternate columns give ranks within rows.
Row
1
2
3
4
Fifi
38
22
19
16
C31
2
2
1
1
Adelina
41
14
20
18
C32
3
1
2
2
Maria
23
32
28
22
C33
1
3
3
3
Data Display Display for Friedman test. Gives sums of ranks by column.
rs1
rs2
rs3
6.00000
8.00000
10.0000
51
252y0771t 11/28/07
Friedman Test: CCalls versus Soprano blocked by Venue Test of equality of medians blocked
by Venue. High p-value means do not reject equality of medians.
S = 2.00 DF = 2 P = 0.368
Sum of
Soprano N Est Median
Ranks
Adelina 4
22.000
8.0
Fifi
4
20.500
6.0
Maria
4
28.000
10.0
Grand median = 23.500
Data Display Display for Levene test. Original data.
Row
1
2
3
4
Fifi_1
38
22
19
16
Adelina_1
41
14
20
18
Maria_1
23
32
28
22
Data Display Display for Levene test. Column medians.
K41
K42
K43
20.5000
19.0000
25.5000
Data Display Display for Levene test. Columns less medians.
Row
1
2
3
4
Fifi_1
17.5
1.5
-1.5
-4.5
Adelina_1
22
-5
1
-1
Maria_1
-2.5
6.5
2.5
-3.5
Data Display Display for Levene test. Absolute value of Columns less medians.
Row
1
2
3
4
Fifi_1
17.5
1.5
1.5
4.5
Adelina_1
22
5
1
1
Maria_1
2.5
6.5
2.5
3.5
One-way ANOVA: Fifi_1, Adelina_1, Maria_1 Test of equality of variances. High p-value means do
not reject equality of variances.
Source DF
SS
MS
F
P
Factor
2
26.0 13.0 0.24 0.791
Error
9 486.3 54.0
Total
11 512.3
S = 7.350
R-Sq = 5.08%
R-Sq(adj) = 0.00%
Individual 90% CIs For Mean Based on
Pooled StDev
Level
N
Mean
StDev ------+---------+---------+---------+--Fifi_1
4 6.250
7.632
(-------------*------------)
Adelina_1 4 7.250 10.012
(-------------*------------)
Maria_1
4 3.750
1.893 (-------------*------------)
------+---------+---------+---------+--0.0
5.0
10.0
15.0
Pooled StDev = 7.350
52
252y0771t 11/28/07
Test for Equal Variances: CCalls versus Soprano Test of equality of variances. High p-value
means do not reject equality of variances.
90% Bonferroni confidence intervals for standard deviations
Soprano N
Lower
StDev
Upper
Adelina 4 6.54717 12.0934 51.8864
Fifi 4 5.31136
9.8107 42.0927
Maria 4 2.51516
4.6458 19.9327
Bartlett's Test (Normal Distribution)
Test statistic = 2.11, p-value = 0.348
Levene's Test (Any Continuous Distribution)
Test statistic = 0.24, p-value = 0.791
Test for Equal Variances: CCalls versus Soprano
53
252y0771t 11/28/07
Version 3
MTB > WOpen "C:\Documents and Settings\RBOVE\My Documents\Minitab\252x07712a4.MTW".
Retrieving worksheet from file: 'C:\Documents and Settings\RBOVE\My
Documents\Minitab\252x0771-2a4.MTW'
Worksheet was saved on Thu Nov 29 2007
Results for: 252x0771-2a4.MTW
MTB > Execute "C:\Documents and Settings\RBOVE\My
Documents\Minitab\ANOVA252y077.mtb" 1.
Executing from file: C:\Documents and Settings\RBOVE\My
Documents\Minitab\ANOVA252y077.mtb
Data Display
Row
1
2
3
4
Fifi
39
22
19
16
Adelina
42
14
20
18
Maria
24
32
28
22
Data Display Display for ANOVA tableau. Includes Total sum, Total sum of squares, grand mean, SST, SSB =
SSC, Sums of squared row and column means.
SumAll
296.000
SumAllsq
8174.00
grandM
24.6667
SST
872.667
SSC
20.6667
SSR
456.667
SColMsq
1830.50
SRowMsq
2586.00
* NOTE * One or more variables are undefined.
Data Display Display for ANOVA tableau. Includes Row sums, Column sums, Row and Column sum of squares,
Row and column means and row and column means squared.
Row RowS ColS RowSS ColSS RowMean ColMean
1
105
96
3861
2622 35.0000
24.0
2
68
94
1704
2684 22.6667
23.5
3
67
106
1545
2868 22.3333
26.5
4
56
1064
18.6667
ColMsq
576.00
552.25
702.25
RowMsq
1225.00
513.78
498.78
348.44
One-way ANOVA: Fifi, Adelina, Maria Test of equality of column means. High p-value means do not
reject equality.
Source DF
Factor
2
Error
9
Total
11
S = 9.730
Level
Fifi
Adelina
Maria
N
4
4
4
SS
MS
F
P
20.7 10.3 0.11 0.898
852.0 94.7
872.7
R-Sq = 2.37%
R-Sq(adj) = 0.00%
Mean
24.000
23.500
26.500
StDev
10.296
12.583
4.435
Individual 95% CIs For Mean Based on
Pooled StDev
--+---------+---------+---------+------(--------------*---------------)
(---------------*--------------)
(---------------*---------------)
--+---------+---------+---------+------14.0
21.0
28.0
35.0
Pooled StDev = 9.730
54
252y0771t 11/28/07
Two-way ANOVA: CCalls versus Venue, Soprano Test of equality of row and column means.
High p-value means do not reject equality.
Source
DF
SS
MS
F
P
Venue
3 456.667 152.222 2.31 0.176
Soprano
2
20.667
10.333 0.16 0.858
Error
6 395.333
65.889
Total
11 872.667
S = 8.117
R-Sq = 54.70%
R-Sq(adj) = 16.95%
Individual 95% CIs For Mean Based on
Pooled StDev
Venue
Mean ---+---------+---------+---------+-----Milan
35.0000
(----------*----------)
Naples 22.3333
(----------*-----------)
Rome
18.6667 (-----------*----------)
Venice 22.6667
(-----------*----------)
---+---------+---------+---------+-----10
20
30
40
Individual 95% CIs For Mean Based on
Pooled StDev
Soprano Mean -------+---------+---------+---------+-Adelina 23.5 (---------------*----------------)
Fifi
24.0 (----------------*----------------)
Maria
26.5
(---------------*----------------)
-------+---------+---------+---------+-18.0
24.0
30.0
36.0
Data Display Display for Kruskal-Wallis test. Alternate columns give ranks.
Row
1
2
3
4
Fifi
39
22
19
16
rCCalls_Fifi
11.0
6.5
4.0
2.0
Adelina
42
14
20
18
rCCalls_Adelina
12
1
5
3
Maria
24
32
28
22
rCCalls_Maria
8.0
10.0
9.0
6.5
Data Display Display for Kruskal-Wallis test. Gives sums of column ranks.
rs1
rs2
rs3
23.5000
21.0000
33.5000
Kruskal-Wallis Test: CCalls versus Soprano Test of equality of medians. High p-value means do
not reject equality of medians.
Kruskal-Wallis Test on CCalls
Ave
Soprano
N Median Rank
Z
Adelina
4
19.00
5.3 -0.85
Fifi
4
20.50
5.9 -0.42
Maria
4
26.00
8.4
1.27
Overall 12
6.5
H = 1.68 DF = 2 P = 0.431
H = 1.69 DF = 2 P = 0.430 (adjusted for ties)
* NOTE * One or more small samples
Data Display Display for Friedman test. Alternate columns give ranks within rows.
Row
1
2
3
4
Fifi
39
22
19
16
C31
2
2
1
1
Adelina
42
14
20
18
C32
3
1
2
2
Maria
24
32
28
22
C33
1
3
3
3
Data Display Display for Friedman test. Gives sums of ranks by column.
rs1
rs2
rs3
6.00000
8.00000
10.0000
55
252y0771t 11/28/07
Friedman Test: CCalls versus Soprano blocked by Venue Test of equality of medians blocked
by Venue. High p-value means do not reject equality of medians.
S = 2.00 DF = 2 P = 0.368
Sum of
Soprano N Est Median
Ranks
Adelina 4
22.000
8.0
Fifi
4
20.500
6.0
Maria
4
28.000
10.0
Grand median = 23.500
Data Display Display for Levene test. Original data.
Row
1
2
3
4
Fifi_1
39
22
19
16
Adelina_1
42
14
20
18
Maria_1
24
32
28
22
Data Display Display for Levene test. Column medians.
K41
K42
K43
20.5000
19.0000
26.0000
Data Display Display for Levene test. Columns less medians.
Row
1
2
3
4
Fifi_1
18.5
1.5
-1.5
-4.5
Adelina_1
23
-5
1
-1
Maria_1
-2
6
2
-4
Data Display Display for Levene test. Absolute value of Columns less medians.
Row
1
2
3
4
Fifi_1
18.5
1.5
1.5
4.5
Adelina_1
23
5
1
1
Maria_1
2
6
2
4
One-way ANOVA: Fifi_1, Adelina_1, Maria_1 Test of equality of variances. High p-value means do
not reject equality of variances.
Source DF
SS
MS
F
P
Factor
2
34.7 17.3 0.29 0.756
Error
9 540.0 60.0
Total
11 574.7
S = 7.746
R-Sq = 6.03%
R-Sq(adj) = 0.00%
Level
Fifi_1
Adelina_1
Maria_1
N
4
4
4
Mean
6.500
7.500
3.500
StDev
8.124
10.504
1.915
Individual 90% CIs For Mean Based on
Pooled StDev
-------+---------+---------+---------+-(-------------*-------------)
(-------------*-------------)
(-------------*-------------)
-------+---------+---------+---------+-0.0
5.0
10.0
15.0
Pooled StDev = 7.746
56
252y0771t 11/28/07
Test for Equal Variances: CCalls versus Soprano Test of equality of variances. High p-value
means do not reject equality of variances.
90% Bonferroni confidence intervals for standard deviations
Soprano N
Lower
StDev
Upper
Adelina 4 6.81227 12.5831 53.9874
Fifi 4 5.57389 10.2956 44.1732
Maria 4 2.40088
4.4347 19.0270
Bartlett's Test (Normal Distribution)
Test statistic = 2.47, p-value = 0.291
Levene's Test (Any Continuous Distribution)
Test statistic = 0.29, p-value = 0.756
Test for Equal Variances: CCalls versus Soprano
57
252y0771t 11/28/07
Version 4
MTB > WOpen "C:\Documents and Settings\RBOVE\My Documents\Minitab\252x07712a5.MTW".
Retrieving worksheet from file: 'C:\Documents and Settings\RBOVE\My
Documents\Minitab\252x0771-2a5.MTW'
Worksheet was saved on Thu Nov 29 2007
Results for: 252x0771-2a5.MTW
MTB > Execute "C:\Documents and Settings\RBOVE\My
Documents\Minitab\ANOVA252y077.mtb" 1.
Executing from file: C:\Documents and Settings\RBOVE\My
Documents\Minitab\ANOVA252y077.mtb
Data Display
Row
1
2
3
4
Fifi
40
22
19
16
Adelina
43
14
20
18
Maria
25
32
28
22
Data Display Display for ANOVA tableau. Includes Total sum, Total sum of squares, grand mean, SST, SSB =
SSC, Sums of squared row and column means.
SumAll
299.000
SumAllsq
8387.00
grandM
24.9167
SST
936.917
SSC
20.6667
SSR
520.917
SColMsq
1867.69
SRowMsq
2657.00
* NOTE * One or more variables are undefined.
Data Display Display for ANOVA tableau. Includes Row sums, Column sums, Row and Column sum of squares,
Row and column means and row and column means squared.
Row RowS ColS RowSS ColSS RowMean ColMean
1
108
97
4074
2701 36.0000
24.25
2
68
95
1704
2769 22.6667
23.75
3
67
107
1545
2917 22.3333
26.75
4
56
1064
18.6667
ColMsq
588.063
564.063
715.563
RowMsq
1296.00
513.78
498.78
348.44
One-way ANOVA: Fifi, Adelina, Maria Test of equality of column means. High p-value means do not
reject equality.
Source DF
Factor
2
Error
9
Total
11
S = 10.09
SS
MS
F
P
21
10 0.10 0.905
916 102
937
R-Sq = 2.21%
R-Sq(adj) = 0.00%
Individual 95% CIs For Mean Based on
Pooled StDev
Level
N
Mean StDev
--+---------+---------+---------+------Fifi
4 24.25 10.78
(----------------*---------------)
Adelina 4 23.75 13.07
(---------------*---------------)
Maria
4 26.75
4.27
(---------------*----------------)
--+---------+---------+---------+------14.0
21.0
28.0
35.0
Pooled StDev = 10.09
Two-way ANOVA: CCalls versus Venue, Soprano Test of equality of row and column means.
High p-value means do not reject equality.
Source
DF
SS
MS
F
P
Venue
3 520.917 173.639 2.64 0.144
Soprano
2
20.667
10.333 0.16 0.858
Error
6 395.333
65.889
Total
11 936.917
S = 8.117
R-Sq = 57.80%
R-Sq(adj) = 22.64%
58
252y0771t 11/28/07
Venue
Milan
Naples
Rome
Venice
Mean
36.0000
22.3333
18.6667
22.6667
Soprano
Adelina
Fifi
Maria
Mean
23.75
24.25
26.75
Individual 95% CIs For Mean Based on
Pooled StDev
----+---------+---------+---------+----(---------*---------)
(---------*--------)
(---------*--------)
(---------*--------)
----+---------+---------+---------+----12
24
36
48
Individual 95% CIs For Mean Based on
Pooled StDev
-------+---------+---------+---------+-(----------------*---------------)
(---------------*----------------)
(----------------*---------------)
-------+---------+---------+---------+-18.0
24.0
30.0
36.0
Data Display Display for Kruskal-Wallis test. Alternate columns give ranks.
Row
1
2
3
4
Fifi
40
22
19
16
rCCalls_Fifi
11.0
6.5
4.0
2.0
Adelina
43
14
20
18
rCCalls_Adelina
12
1
5
3
Maria
25
32
28
22
rCCalls_Maria
8.0
10.0
9.0
6.5
Data Display Display for Kruskal-Wallis test. Gives sums of column ranks.
rs1
rs2
rs3
23.5000
21.0000
33.5000
Kruskal-Wallis Test: CCalls versus Soprano Test of equality of medians. High p-value means do
not reject equality of medians.
Kruskal-Wallis Test on CCalls
Ave
Soprano
N Median Rank
Z
Adelina
4
19.00
5.3 -0.85
Fifi
4
20.50
5.9 -0.42
Maria
4
26.50
8.4
1.27
Overall 12
6.5
H = 1.68 DF = 2 P = 0.431
H = 1.69 DF = 2 P = 0.430 (adjusted for ties)
* NOTE * One or more small samples
Data Display Display for Friedman test. Alternate columns give ranks within rows.
Row
1
2
3
4
Fifi
40
22
19
16
C31
2
2
1
1
Adelina
43
14
20
18
C32
3
1
2
2
Maria
25
32
28
22
C33
1
3
3
3
Data Display Display for Friedman test. Gives sums of ranks by column.
rs1
rs2
rs3
6.00000
8.00000
10.0000
59
252y0771t 11/28/07
Friedman Test: CCalls versus Soprano blocked by Venue Test of equality of medians blocked
by Venue. High p-value means do not reject equality of medians.
S = 2.00 DF = 2 P = 0.368
Sum of
Soprano N Est Median
Ranks
Adelina 4
22.000
8.0
Fifi
4
20.500
6.0
Maria
4
28.000
10.0
Grand median = 23.500
Data Display Display for Levene test. Original data.
Row
1
2
3
4
Fifi_1
40
22
19
16
Adelina_1
43
14
20
18
Maria_1
25
32
28
22
Data Display Display for Levene test. Column medians.
K41
K42
K43
20.5000
19.0000
26.5000
Data Display Display for Levene test. Columns less medians.
Row
1
2
3
4
Fifi_1
19.5
1.5
-1.5
-4.5
Adelina_1
24
-5
1
-1
Maria_1
-1.5
5.5
1.5
-4.5
Data Display Display for Levene test. Absolute value of Columns less medians.
Row
1
2
3
4
Fifi_1
19.5
1.5
1.5
4.5
Adelina_1
24
5
1
1
Maria_1
1.5
5.5
1.5
4.5
One-way ANOVA: Fifi_1, Adelina_1, Maria_1 Test of equality of variances. High p-value means do
not reject equality of variances.
Source DF
SS
MS
F
P
Factor
2
44.7 22.3 0.34 0.723
Error
9 598.3 66.5
Total
11 642.9
S = 8.153
R-Sq = 6.95%
R-Sq(adj) = 0.00%
Individual 90% CIs For Mean Based on
Pooled StDev
Level
N
Mean
StDev --------+---------+---------+---------+Fifi_1
4 6.750
8.617
(--------------*-------------)
Adelina_1 4 7.750 10.996
(--------------*-------------)
Maria_1
4 3.250
2.062 (--------------*-------------)
--------+---------+---------+---------+0.0
5.0
10.0
15.0
Pooled StDev = 8.153
60
252y0771t 11/28/07
Test for Equal Variances: CCalls versus Soprano Test of equality of variances. High p-value
means do not reject equality of variances.
90% Bonferroni confidence intervals for standard deviations
Soprano N
Lower
StDev
Upper
Adelina 4 7.07779 13.0735 56.0916
Fifi 4 5.83717 10.7819 46.2597
Maria 4 2.31280
4.2720 18.3289
Bartlett's Test (Normal Distribution)
Test statistic = 2.79, p-value = 0.248
Levene's Test (Any Continuous Distribution)
Test statistic = 0.34, p-value = 0.723
Test for Equal Variances: CCalls versus Soprano
61
252y0771t 11/28/07
Version 5
MTB > WOpen "C:\Documents and Settings\RBOVE\My Documents\Minitab\252x07712a6.MTW".
Retrieving worksheet from file: 'C:\Documents and Settings\RBOVE\My
Documents\Minitab\252x0771-2a6.MTW'
Worksheet was saved on Thu Nov 29 2007
Results for: 252x0771-2a6.MTW
MTB > Execute "C:\Documents and Settings\RBOVE\My
Documents\Minitab\ANOVA252y077.mtb" 1.
Executing from file: C:\Documents and Settings\RBOVE\My
Documents\Minitab\ANOVA252y077.mtb
Data Display
Row
1
2
3
4
Fifi
41
22
19
16
Adelina
44
14
20
18
Maria
26
32
28
22
Data Display Display for ANOVA tableau. Includes Total sum, Total sum of squares, grand mean, SST, SSB =
SSC, Sums of squared row and column means.
SumAll
302.000
SumAllsq
8606.00
grandM
25.1667
SST
1005.67
SSC
20.6667
SSR
589.667
SColMsq
1905.25
SRowMsq
2730.00
* NOTE * One or more variables are undefined.
Data Display Display for ANOVA tableau. Includes Row sums, Column sums, Row and Column sum of squares,
Row and column means and row and column means squared.
Row RowS ColS RowSS ColSS RowMean ColMean
1
111
98
4293
2782 37.0000
24.5
2
68
96
1704
2856 22.6667
24.0
3
67
108
1545
2968 22.3333
27.0
4
56
1064
18.6667
ColMsq
600.25
576.00
729.00
RowMsq
1369.00
513.78
498.78
348.44
One-way ANOVA: Fifi, Adelina, Maria Test of equality of column means. High p-value means do not
reject equality.
Source DF
Factor
2
Error
9
Total
11
S = 10.46
SS
MS
F
P
21
10 0.09 0.911
985 109
1006
R-Sq = 2.06%
R-Sq(adj) = 0.00%
Individual 95% CIs For Mean Based on
Pooled StDev
Level
N
Mean StDev ---+---------+---------+---------+-----Fifi
4 24.50 11.27
(----------------*----------------)
Adelina 4 24.00 13.56 (----------------*----------------)
Maria
4 27.00
4.16
(----------------*---------------)
---+---------+---------+---------+-----14.0
21.0
28.0
35.0
Pooled StDev = 10.46
62
252y0771t 11/28/07
Two-way ANOVA: CCalls versus Venue, Soprano Test of equality of row and column means.
High p-value means do not reject equality.
Source
DF
SS
MS
F
P
Venue
3
589.67 196.556 2.98 0.118
Soprano
2
20.67
10.333 0.16 0.858
Error
6
395.33
65.889
Total
11 1005.67
S = 8.117
R-Sq = 60.69%
R-Sq(adj) = 27.93%
Individual 95% CIs For Mean Based on
Pooled StDev
Venue
Mean ----+---------+---------+---------+----Milan
37.0000
(---------*--------)
Naples 22.3333
(---------*--------)
Rome
18.6667 (---------*--------)
Venice 22.6667
(---------*--------)
----+---------+---------+---------+----12
24
36
48
Soprano
Adelina
Fifi
Maria
Mean
24.0
24.5
27.0
Individual 95% CIs For Mean Based on
Pooled StDev
-------+---------+---------+---------+-(----------------*----------------)
(----------------*---------------)
(----------------*----------------)
-------+---------+---------+---------+-18.0
24.0
30.0
36.0
Data Display Display for Kruskal-Wallis test. Alternate columns give ranks.
Row
1
2
3
4
Fifi
41
22
19
16
rCCalls_Fifi
11.0
6.5
4.0
2.0
Adelina
44
14
20
18
rCCalls_Adelina
12
1
5
3
Maria
26
32
28
22
rCCalls_Maria
8.0
10.0
9.0
6.5
Data Display Display for Kruskal-Wallis test. Gives sums of column ranks.
rs1
rs2
rs3
23.5000
21.0000
33.5000
Kruskal-Wallis Test: CCalls versus Soprano Test of equality of medians. High p-value means do
not reject equality of medians.
Kruskal-Wallis Test on CCalls
Ave
Soprano
N Median Rank
Z
Adelina
4
19.00
5.3 -0.85
Fifi
4
20.50
5.9 -0.42
Maria
4
27.00
8.4
1.27
Overall 12
6.5
H = 1.68 DF = 2 P = 0.431
H = 1.69 DF = 2 P = 0.430 (adjusted for ties)
* NOTE * One or more small samples
Data Display Display for Friedman test. Alternate columns give ranks within rows.
Row
1
2
3
4
Fifi
41
22
19
16
C31
2
2
1
1
Adelina
44
14
20
18
C32
3
1
2
2
Maria
26
32
28
22
C33
1
3
3
3
Data Display Display for Friedman test. Gives sums of ranks by column.
rs1
rs2
rs3
6.00000
8.00000
10.0000
63
252y0771t 11/28/07
Friedman Test: CCalls versus Soprano blocked by Venue Test of equality of medians blocked
by Venue. High p-value means do not reject equality of medians.
S = 2.00 DF = 2 P = 0.368
Sum of
Soprano N Est Median
Ranks
Adelina 4
22.000
8.0
Fifi
4
20.500
6.0
Maria
4
28.000
10.0
Grand median = 23.500
Data Display Display for Levene test. Original data.
Row
1
2
3
4
Fifi_1
41
22
19
16
Adelina_1
44
14
20
18
Maria_1
26
32
28
22
Data Display Display for Levene test. Column medians.
K41
K42
K43
20.5000
19.0000
27.0000
Data Display Display for Levene test. Columns less medians.
Row
1
2
3
4
Fifi_1
20.5
1.5
-1.5
-4.5
Adelina_1
25
-5
1
-1
Maria_1
-1
5
1
-5
Data Display Display for Levene test. Absolute value of Columns less medians.
Row
1
2
3
4
Fifi_1
20.5
1.5
1.5
4.5
Adelina_1
25
5
1
1
Maria_1
1
5
1
5
One-way ANOVA: Fifi_1, Adelina_1, Maria_1 Test of equality of variances. High p-value means do
not reject equality of variances.
Source DF
SS
MS
F
P
Factor
2
56.0 28.0 0.38 0.694
Error
9 661.0 73.4
Total
11 717.0
S = 8.570
R-Sq = 7.81%
R-Sq(adj) = 0.00%
Individual 90% CIs For Mean Based on
Pooled StDev
Level
N
Mean
StDev --------+---------+---------+---------+Fifi_1
4 7.000
9.110
(------------*------------)
Adelina_1 4 8.000 11.489
(------------*------------)
Maria_1
4 3.000
2.309 (------------*------------)
--------+---------+---------+---------+0.0
6.0
12.0
18.0
Pooled StDev = 8.570
64
252y0771t 11/28/07
Test for Equal Variances: CCalls versus Soprano Test of equality of variances. High p-value
means do not reject equality of variances.
90% Bonferroni confidence intervals for standard deviations
Soprano N
Lower
StDev
Upper
Adelina 4 7.34369 13.5647 58.1989
Fifi 4 6.10109 11.2694 48.3513
Maria 4 2.25396
4.1633 17.8627
Bartlett's Test (Normal Distribution)
Test statistic = 3.07, p-value = 0.216
Levene's Test (Any Continuous Distribution)
Test statistic = 0.38, p-value = 0.694
Test for Equal Variances: CCalls versus Soprano
65
252y0771t 11/28/07
Version 6
MTB > WOpen "C:\Documents and Settings\RBOVE\My Documents\Minitab\252x07712a7.MTW".
Retrieving worksheet from file: 'C:\Documents and Settings\RBOVE\My
Documents\Minitab\252x0771-2a7.MTW'
Worksheet was saved on Thu Nov 29 2007
Results for: 252x0771-2a7.MTW
MTB > Execute "C:\Documents and Settings\RBOVE\My
Documents\Minitab\ANOVA252y077.mtb" 1.
Executing from file: C:\Documents and Settings\RBOVE\My
Documents\Minitab\ANOVA252y077.mtb
Data Display
Row
1
2
3
4
Fifi
42
22
19
16
Adelina
45
14
20
18
Maria
27
32
28
22
Data Display Display for ANOVA tableau. Includes Total sum, Total sum of squares, grand mean, SST, SSB =
SSC, Sums of squared row and column means.
SumAll
305.000
SumAllsq
8831.00
grandM
25.4167
SST
1078.92
SSC
20.6667
SSR
662.917
SColMsq
1943.19
SRowMsq
2805.00
* NOTE * One or more variables are undefined.
Data Display Display for ANOVA tableau. Includes Row sums, Column sums, Row and Column sum of
squares, Row and column means and row and column means squared.
Row RowS ColS RowSS ColSS RowMean ColMean
1
114
99
4518
2865 38.0000
24.75
2
68
97
1704
2945 22.6667
24.25
3
67
109
1545
3021 22.3333
27.25
4
56
1064
18.6667
ColMsq
612.563
588.063
742.563
RowMsq
1444.00
513.78
498.78
348.44
One-way ANOVA: Fifi, Adelina, Maria Test of equality of column means. High p-value means do not
reject equality.
Source DF
Factor
2
Error
9
Total
11
S = 10.84
SS
MS
F
P
21
10 0.09 0.917
1058 118
1079
R-Sq = 1.92%
R-Sq(adj) = 0.00%
Individual 95% CIs For Mean Based on
Pooled StDev
Level
N
Mean StDev ---+---------+---------+---------+-----Fifi
4 24.75 11.76
(----------------*-----------------)
Adelina 4 24.25 14.06 (-----------------*----------------)
Maria
4 27.25
4.11
(-----------------*----------------)
---+---------+---------+---------+-----14.0
21.0
28.0
35.0
Pooled StDev = 10.84
66
252y0771t 11/28/07
Two-way ANOVA: CCalls versus Venue, Soprano Test of equality of row and column means.
High p-value means do not reject equality. But note that the p-value for Venue is now below 10%. This means that we
reject the null hypothesis and conclude that row means differ.
Source
DF
SS
MS
F
P
Venue
3
662.92 220.972 3.35 0.097
Soprano
2
20.67
10.333 0.16 0.858
Error
6
395.33
65.889
Total
11 1078.92
S = 8.117
R-Sq = 63.36%
R-Sq(adj) = 32.82%
Venue
Milan
Naples
Rome
Venice
Mean
38.0000
22.3333
18.6667
22.6667
Soprano
Adelina
Fifi
Maria
Mean
24.25
24.75
27.25
Individual 95% CIs For Mean Based on
Pooled StDev
----+---------+---------+---------+----(---------*--------)
(---------*--------)
(---------*--------)
(---------*--------)
----+---------+---------+---------+----12
24
36
48
Individual 95% CIs For Mean Based on
Pooled StDev
------+---------+---------+---------+--(---------------*----------------)
(---------------*----------------)
(---------------*----------------)
------+---------+---------+---------+--18.0
24.0
30.0
36.0
Data Display Display for Kruskal-Wallis test. Alternate columns give ranks.
Row
1
2
3
4
Fifi
42
22
19
16
rCCalls_Fifi
11.0
6.5
4.0
2.0
Adelina
45
14
20
18
rCCalls_Adelina
12
1
5
3
Maria
27
32
28
22
rCCalls_Maria
8.0
10.0
9.0
6.5
Data Display Display for Kruskal-Wallis test. Gives sums of column ranks.
rs1
rs2
rs3
23.5000
21.0000
33.5000
Kruskal-Wallis Test: CCalls versus Soprano Test of equality of medians. High p-value means do
not reject equality of medians.
Kruskal-Wallis Test on CCalls
Ave
Soprano
N Median Rank
Z
Adelina
4
19.00
5.3 -0.85
Fifi
4
20.50
5.9 -0.42
Maria
4
27.50
8.4
1.27
Overall 12
6.5
H = 1.68 DF = 2 P = 0.431
H = 1.69 DF = 2 P = 0.430 (adjusted for ties)
* NOTE * One or more small samples
Data Display Display for Friedman test. Alternate columns give ranks within rows.
Row
1
2
3
4
Fifi
42
22
19
16
C31
2
2
1
1
Adelina
45
14
20
18
C32
3
1
2
2
Maria
27
32
28
22
C33
1
3
3
3
67
252y0771t 11/28/07
Data Display Display for Friedman test. Gives sums of ranks by column.
rs1
rs2
rs3
6.00000
8.00000
10.0000
Friedman Test: CCalls versus Soprano blocked by Venue Test of equality of medians
blocked by Venue. High p-value means do not reject equality of medians.
S = 2.00 DF = 2 P = 0.368
Sum of
Soprano N Est Median
Ranks
Adelina 4
22.000
8.0
Fifi
4
20.500
6.0
Maria
4
28.000
10.0
Grand median = 23.500
Data Display Display for Levene test. Original data.
Row
1
2
3
4
Fifi_1
42
22
19
16
Adelina_1
45
14
20
18
Maria_1
27
32
28
22
Data Display Display for Levene test. Column medians.
K41
K42
K43
20.5000
19.0000
27.5000
Data Display Display for Levene test. Columns less medians.
Row
1
2
3
4
Fifi_1
21.5
1.5
-1.5
-4.5
Adelina_1
26
-5
1
-1
Maria_1
-0.5
4.5
0.5
-5.5
Data Display Display for Levene test. Absolute value of Columns less medians.
Row
1
2
3
4
Fifi_1
21.5
1.5
1.5
4.5
Adelina_1
26
5
1
1
Maria_1
0.5
4.5
0.5
5.5
One-way ANOVA: Fifi_1, Adelina_1, Maria_1 Test of equality of variances. High p-value means do
not reject equality of variances.
Source DF
SS
MS
F
P
Factor
2
68.7 34.3 0.42 0.667
Error
9 728.3 80.9
Total
11 796.9
S = 8.995
R-Sq = 8.62%
R-Sq(adj) = 0.00%
Level
Fifi_1
Adelina_1
Maria_1
N
4
4
4
Mean
7.250
8.250
2.750
StDev
9.605
11.983
2.630
Individual 90% CIs For Mean Based on
Pooled StDev
---------+---------+---------+---------+
(-------------*-------------)
(-------------*------------)
(-------------*------------)
---------+---------+---------+---------+
0.0
6.0
12.0
18.0
Pooled StDev = 8.995
68
252y0771t 11/28/07
Test for Equal Variances: CCalls versus Soprano Test of equality of variances. High p-value
means do not reject equality of variances.
90% Bonferroni confidence intervals for standard deviations
Soprano N
Lower
StDev
Upper
Adelina 4 7.60993 14.0564 60.3089
Fifi 4 6.36558 11.7580 50.4474
Maria 4 2.22671
4.1130 17.6467
Bartlett's Test (Normal Distribution)
Test statistic = 3.29, p-value = 0.193
Levene's Test (Any Continuous Distribution)
Test statistic = 0.42, p-value = 0.667
Test for Equal Variances: CCalls versus Soprano
69
252y0771t 11/28/07
Version 7
MTB > WOpen "C:\Documents and Settings\RBOVE\My Documents\Minitab\252x07712a8.MTW".
Retrieving worksheet from file: 'C:\Documents and Settings\RBOVE\My
Documents\Minitab\252x0771-2a8.MTW'
Worksheet was saved on Thu Nov 29 2007
Results for: 252x0771-2a8.MTW
MTB > Execute "C:\Documents and Settings\RBOVE\My
Documents\Minitab\ANOVA252y077.mtb" 1.
Executing from file: C:\Documents and Settings\RBOVE\My
Documents\Minitab\ANOVA252y077.mtb
Data Display
Row
1
2
3
4
Fifi
43
22
19
16
Adelina
46
14
20
18
Maria
28
32
28
22
Data Display Display for ANOVA tableau. Includes Total sum, Total sum of squares, grand mean, SST, SSB =
SSC, Sums of squared row and column means.
SumAll
308.000
SumAllsq
9062.00
grandM
25.6667
SST
1156.67
SSC
20.6667
SSR
740.667
SColMsq
1981.50
SRowMsq
2882.00
* NOTE * One or more variables are undefined.
Data Display Display for ANOVA tableau. Includes Row sums, Column sums, Row and Column sum of
squares, Row and column means and row and column means squared.
Row RowS ColS RowSS ColSS RowMean ColMean
1
117
100
4749
2950 39.0000
25.0
2
68
98
1704
3036 22.6667
24.5
3
67
110
1545
3076 22.3333
27.5
4
56
1064
18.6667
ColMsq
625.00
600.25
756.25
RowMsq
1521.00
513.78
498.78
348.44
One-way ANOVA: Fifi, Adelina, Maria Test of equality of column means. High p-value means do not
reject equality.
Source DF
Factor
2
Error
9
Total
11
S = 11.23
SS
MS
F
P
21
10 0.08 0.922
1136 126
1157
R-Sq = 1.79%
R-Sq(adj) = 0.00%
Individual 95% CIs For Mean Based on
Pooled StDev
Level
N
Mean StDev -----+---------+---------+---------+---Fifi
4 25.00 12.25 (---------------*---------------)
Adelina 4 24.50 14.55 (---------------*---------------)
Maria
4 27.50
4.12
(---------------*---------------)
-----+---------+---------+---------+---16.0
24.0
32.0
40.0
Pooled StDev = 11.23
70
252y0771t 11/28/07
Two-way ANOVA: CCalls versus Venue, Soprano Test of equality of row and column means.
High p-value means do not reject equality. But note that the p-value for Venue is now below 10%. This means that we
reject the null hypothesis and conclude that row means differ.
Source
DF
SS
MS
F
P
Venue
3
740.67 246.889 3.75 0.079
Soprano
2
20.67
10.333 0.16 0.858
Error
6
395.33
65.889
Total
11 1156.67
S = 8.117
R-Sq = 65.82%
R-Sq(adj) = 37.34%
Individual 95% CIs For Mean Based on
Pooled StDev
Venue
Mean ----+---------+---------+---------+----Milan
39.0000
(---------*--------)
Naples 22.3333
(---------*--------)
Rome
18.6667 (---------*--------)
Venice 22.6667
(---------*--------)
----+---------+---------+---------+----12
24
36
48
Individual 95% CIs For Mean Based on
Pooled StDev
Soprano Mean ------+---------+---------+---------+--Adelina 24.5 (----------------*---------------)
Fifi
25.0
(----------------*---------------)
Maria
27.5
(----------------*---------------)
------+---------+---------+---------+--18.0
24.0
30.0
36.0
Data Display
Row
1
2
3
4
Fifi
43
22
19
16
Display for Kruskal-Wallis test. Alternate columns give ranks.
rCCalls_Fifi Adelina rCCalls_Adelina Maria rCCalls_Maria
11.0
46
12
28
8.5
6.5
14
1
32
10.0
4.0
20
5
28
8.5
2.0
18
3
22
6.5
Data Display Display for Kruskal-Wallis test. Gives sums of column ranks.
rs1
rs2
rs3
23.5000
21.0000
33.5000
Kruskal-Wallis Test: CCalls versus Soprano Test of equality of medians. High p-value means do
not reject equality of medians.
Kruskal-Wallis Test on CCalls
Ave
Soprano
N Median Rank
Z
Adelina
4
19.00
5.3 -0.85
Fifi
4
20.50
5.9 -0.42
Maria
4
28.00
8.4
1.27
Overall 12
6.5
H = 1.68 DF = 2 P = 0.431
H = 1.69 DF = 2 P = 0.429 (adjusted for ties)
* NOTE * One or more small samples
Data Display Display for Friedman test. Alternate columns give ranks within rows.
Row
1
2
3
4
Fifi
43
22
19
16
C31
2
2
1
1
Adelina
46
14
20
18
C32
3
1
2
2
Maria
28
32
28
22
C33
1
3
3
3
Data Display Display for Friedman test. Gives sums of ranks by column.
rs1
rs2
rs3
6.00000
8.00000
10.0000
71
252y0771t 11/28/07
Friedman Test: CCalls versus Soprano blocked by Venue Test of equality of medians blocked
by Venue. High p-value means do not reject equality of medians.
S = 2.00 DF = 2 P = 0.368
Sum of
Soprano N Est Median
Ranks
Adelina 4
22.000
8.0
Fifi
4
20.500
6.0
Maria
4
28.000
10.0
Grand median = 23.500
Data Display Display for Levene test. Original data.
Row
1
2
3
4
Fifi_1
43
22
19
16
Adelina_1
46
14
20
18
Maria_1
28
32
28
22
Data Display Display for Levene test. Column medians.
K41
K42
K43
20.5000
19.0000
28.0000
Data Display Display for Levene test. Columns less medians.
Row
1
2
3
4
Fifi_1
22.5
1.5
-1.5
-4.5
Adelina_1
27
-5
1
-1
Maria_1
0
4
0
-6
Data Display Display for Levene test. Absolute value of Columns less medians.
Row
1
2
3
4
Fifi_1
22.5
1.5
1.5
4.5
Adelina_1
27
5
1
1
Maria_1
0
4
0
6
One-way ANOVA: Fifi_1, Adelina_1, Maria_1 Test of equality of variances. High p-value means do
not reject equality of variances.
Source DF
SS
MS
F
P
Factor
2
82.7 41.3 0.46 0.642
Error
9 800.0 88.9
Total
11 882.7
S = 9.428
R-Sq = 9.37%
R-Sq(adj) = 0.00%
Individual 90% CIs For Mean Based on Pooled StDev
Level
N
Mean
StDev
+---------+---------+---------+--------Fifi_1
4 7.500 10.100
(--------------*-------------)
Adelina_1 4 8.500 12.477
(-------------*--------------)
Maria_1
4 2.500
3.000
(-------------*--------------)
+---------+---------+---------+---------6.0
0.0
6.0
12.0
Pooled StDev = 9.428
72
252y0771t 11/28/07
Test for Equal Variances: CCalls versus Soprano Test of equality of variances. High p-value
means do not reject equality of variances.
90% Bonferroni confidence intervals for standard deviations
Soprano N
Lower
StDev
Upper
Adelina 4 7.87648 14.5488 62.4212
Fifi 4 6.63058 12.2474 52.5474
Maria 4 2.23218
4.1231 17.6901
Bartlett's Test (Normal Distribution)
Test statistic = 3.44, p-value = 0.179
Levene's Test (Any Continuous Distribution)
Test statistic = 0.46, p-value = 0.642
Test for Equal Variances: CCalls versus Soprano
73
252y0771t 11/28/07
Version 8
MTB > WOpen "C:\Documents and Settings\RBOVE\My Documents\Minitab\252x07712a9.MTW".
Retrieving worksheet from file: 'C:\Documents and Settings\RBOVE\My
Documents\Minitab\252x0771-2a9.MTW'
Worksheet was saved on Thu Nov 29 2007
Results for: 252x0771-2a9.MTW
MTB > Execute "C:\Documents and Settings\RBOVE\My
Documents\Minitab\ANOVA252y077.mtb" 1.
Executing from file: C:\Documents and Settings\RBOVE\My
Documents\Minitab\ANOVA252y077.mtb
Data Display
Row
1
2
3
4
Fifi
44
22
19
16
Adelina
47
14
20
18
Maria
29
32
28
22
Data Display Display for ANOVA tableau. Includes Total sum, Total sum of squares, grand mean, SST, SSB =
SSC, Sums of squared row and column means.
SumAll
311.000
SumAllsq
9299.00
grandM
25.9167
SST
1238.92
SSC
20.6667
SSR
822.917
SColMsq
2020.19
SRowMsq
2961.00
* NOTE * One or more variables are undefined.
Data Display Display for ANOVA tableau. Includes Row sums, Column sums, Row and Column sum of squares,
Row and column means and row and column means squared.
Row RowS ColS RowSS ColSS RowMean ColMean
1
120
101
4986
3037 40.0000
25.25
2
68
99
1704
3129 22.6667
24.75
3
67
111
1545
3133 22.3333
27.75
4
56
1064
18.6667
ColMsq
637.563
612.563
770.063
RowMsq
1600.00
513.78
498.78
348.44
One-way ANOVA: Fifi, Adelina, Maria Test of equality of column means. High p-value means do not
reject equality.
Source DF
Factor
2
Error
9
Total
11
S = 11.63
Level
Fifi
Adelina
Maria
N
4
4
4
SS
MS
F
P
21
10 0.08 0.927
1218 135
1239
R-Sq = 1.67%
R-Sq(adj) = 0.00%
Mean
25.25
24.75
27.75
StDev
12.74
15.04
4.19
Individual 95% CIs For Mean Based on
Pooled StDev
------+---------+---------+---------+--(----------------*---------------)
(----------------*---------------)
(----------------*---------------)
------+---------+---------+---------+--16.0
24.0
32.0
40.0
Pooled StDev = 11.63
74
252y0771t 11/28/07
Two-way ANOVA: CCalls versus Venue, Soprano Test of equality of row and column means.
High p-value means do not reject equality. But note that the p-value for Venue is now below 10%. This means that we
reject the null hypothesis and conclude that row means differ.
Source
DF
SS
MS
F
P
Venue
3
822.92 274.306 4.16 0.065
Soprano
2
20.67
10.333 0.16 0.858
Error
6
395.33
65.889
Total
11 1238.92
S = 8.117
R-Sq = 68.09%
R-Sq(adj) = 41.50%
Individual 95% CIs For Mean Based on
Pooled StDev
Venue
Mean ----+---------+---------+---------+----Milan
40.0000
(--------*---------)
Naples 22.3333
(---------*--------)
Rome
18.6667 (---------*--------)
Venice 22.6667
(---------*--------)
----+---------+---------+---------+----12
24
36
48
Soprano
Adelina
Fifi
Maria
Mean
24.75
25.25
27.75
Individual 95% CIs For Mean Based on
Pooled StDev
-----+---------+---------+---------+---(---------------*----------------)
(---------------*----------------)
(---------------*----------------)
-----+---------+---------+---------+---18.0
24.0
30.0
36.0
Data Display Display for Kruskal-Wallis test. Alternate columns give ranks.
Row
1
2
3
4
Fifi
44
22
19
16
rCCalls_Fifi
11.0
6.5
4.0
2.0
Adelina
47
14
20
18
rCCalls_Adelina
12
1
5
3
Maria
29
32
28
22
rCCalls_Maria
9.0
10.0
8.0
6.5
Data Display Display for Kruskal-Wallis test. Gives sums of column ranks.
rs1
rs2
rs3
23.5000
21.0000
33.5000
Kruskal-Wallis Test: CCalls versus Soprano Test of equality of medians. High p-value means do
not reject equality of medians.
Kruskal-Wallis Test on CCalls
Ave
Soprano
N Median Rank
Z
Adelina
4
19.00
5.3 -0.85
Fifi
4
20.50
5.9 -0.42
Maria
4
28.50
8.4
1.27
Overall 12
6.5
H = 1.68 DF = 2 P = 0.431
H = 1.69 DF = 2 P = 0.430 (adjusted for ties)
* NOTE * One or more small samples
Data Display Display for Friedman test. Alternate columns give ranks within rows.
Row
1
2
3
4
Fifi
44
22
19
16
C31
2
2
1
1
Adelina
47
14
20
18
C32
3
1
2
2
Maria
29
32
28
22
C33
1
3
3
3
Data Display Display for Friedman test. Gives sums of ranks by column.
rs1
rs2
rs3
6.00000
8.00000
10.0000
75
252y0771t 11/28/07
Friedman Test: CCalls versus Soprano blocked by Venue Test of equality of medians
blocked by Venue. High p-value means do not reject equality of medians.
S = 2.00 DF = 2 P = 0.368
Sum of
Soprano N Est Median
Ranks
Adelina 4
22.000
8.0
Fifi
4
20.500
6.0
Maria
4
28.000
10.0
Grand median = 23.500
Data Display Display for Levene test. Original data.
Row
1
2
3
4
Fifi_1
44
22
19
16
Adelina_1
47
14
20
18
Maria_1
29
32
28
22
Data Display Display for Levene test. Column medians.
K41
K42
K43
20.5000
19.0000
28.5000
Data Display Display for Levene test. Columns less medians.
Row
1
2
3
4
Fifi_1
23.5
1.5
-1.5
-4.5
Adelina_1
28
-5
1
-1
Maria_1
0.5
3.5
-0.5
-6.5
Data Display Display for Levene test. Absolute value of Columns less medians.
Row
1
2
3
4
Fifi_1
23.5
1.5
1.5
4.5
Adelina_1
28
5
1
1
Maria_1
0.5
3.5
0.5
6.5
One-way ANOVA: Fifi_1, Adelina_1, Maria_1 Test of equality of variances. High p-value means do
not reject equality of variances.
Source DF
SS
MS
F
P
Factor
2
82.7 41.3 0.43 0.664
Error
9 866.2 96.2
Total
11 948.9
S = 9.811
R-Sq = 8.71%
R-Sq(adj) = 0.00%
Individual 90% CIs For Mean Based on Pooled StDev
Level
N
Mean
StDev
+---------+---------+---------+--------Fifi_1
4 7.750 10.595
(--------------*--------------)
Adelina_1 4 8.750 12.971
(--------------*--------------)
Maria_1
4 2.750
2.872
(--------------*--------------)
+---------+---------+---------+---------6.0
0.0
6.0
12.0
Pooled StDev = 9.811
76
252y0771t 11/28/07
Test for Equal Variances: CCalls versus Soprano Test of equality of variances. High p-value
means do not reject equality of variances.
90% Bonferroni confidence intervals for standard deviations
Soprano N
Lower
StDev
Upper
Adelina 4 8.14329 15.0416 64.5357
Fifi 4 6.89601 12.7377 54.6510
Maria 4 2.27016
4.1932 17.9911
Bartlett's Test (Normal Distribution)
Test statistic = 3.52, p-value = 0.172
Levene's Test (Any Continuous Distribution)
Test statistic = 0.43, p-value = 0.664
Test for Equal Variances: CCalls versus Soprano
77
252y0771t 11/28/07
Version 9
MTB > WOpen "C:\Documents and Settings\RBOVE\My Documents\Minitab\252x07712a10.MTW".
Retrieving worksheet from file: 'C:\Documents and Settings\RBOVE\My
Documents\Minitab\252x0771-2a10.MTW'
Worksheet was saved on Thu Nov 29 2007
Results for: 252x0771-2a10.MTW
MTB > Execute "C:\Documents and Settings\RBOVE\My
Documents\Minitab\ANOVA252y077.mtb" 1.
Executing from file: C:\Documents and Settings\RBOVE\My
Documents\Minitab\ANOVA252y077.mtb
Data Display
Row
1
2
3
4
Fifi
45
22
19
16
Adelina
48
14
20
18
Maria
30
32
28
22
Data Display Display for ANOVA tableau. Includes Total sum, Total sum of squares, grand mean, SST, SSB =
SSC, Sums of squared row and column means.
SumAll
314.000
SumAllsq
9542.00
grandM
26.1667
SST
1325.67
SSC
20.6667
SSR
909.667
SColMsq
2059.25
SRowMsq
3042.00
* NOTE * One or more variables are undefined.
Data Display Display for ANOVA tableau. Includes Row sums, Column sums, Row and Column sum of
squares, Row and column means and row and column means squared.
Row RowS ColS RowSS ColSS RowMean ColMean
1
123
102
5229
3126 41.0000
25.5
2
68
100
1704
3224 22.6667
25.0
3
67
112
1545
3192 22.3333
28.0
4
56
1064
18.6667
ColMsq
650.25
625.00
784.00
RowMsq
1681.00
513.78
498.78
348.44
One-way ANOVA: Fifi, Adelina, Maria Test of equality of column means. High p-value means do not
reject equality.
Source DF
Factor
2
Error
9
Total
11
S = 12.04
SS
MS
F
P
21
10 0.07 0.932
1305 145
1326
R-Sq = 1.56%
R-Sq(adj) = 0.00%
Individual 95% CIs For Mean Based on
Pooled StDev
Level
N
Mean StDev ------+---------+---------+---------+--Fifi
4 25.50 13.23
(----------------*----------------)
Adelina 4 25.00 15.53 (----------------*----------------)
Maria
4 28.00
4.32
(----------------*----------------)
------+---------+---------+---------+--16.0
24.0
32.0
40.0
Pooled StDev = 12.04
78
252y0771t 11/28/07
Two-way ANOVA: CCalls versus Venue, Soprano Test of equality of row and column means.
High p-value means do not reject equality. But note that the p-value for Venue is now below 10%. This means that we
reject the null hypothesis and conclude that row means differ.
Source
DF
SS
MS
F
P
Venue
3
909.67 303.222 4.60 0.053
Soprano
2
20.67
10.333 0.16 0.858
Error
6
395.33
65.889
Total
11 1325.67
S = 8.117
R-Sq = 70.18%
R-Sq(adj) = 45.33%
Individual 95% CIs For Mean Based on
Pooled StDev
Venue
Mean ----+---------+---------+---------+----Milan
41.0000
(--------*---------)
Naples 22.3333
(---------*--------)
Rome
18.6667 (---------*--------)
Venice 22.6667
(---------*--------)
----+---------+---------+---------+----12
24
36
48
Individual 95% CIs For Mean Based on
Pooled StDev
Soprano Mean -----+---------+---------+---------+---Adelina 25.0 (----------------*---------------)
Fifi
25.5
(----------------*---------------)
Maria
28.0
(----------------*---------------)
-----+---------+---------+---------+---18.0
24.0
30.0
36.0
Data Display Display for Kruskal-Wallis test. Alternate columns give ranks.
Row
1
2
3
4
Fifi
45
22
19
16
rCCalls_Fifi
11.0
6.5
4.0
2.0
Adelina
48
14
20
18
rCCalls_Adelina
12
1
5
3
Maria
30
32
28
22
rCCalls_Maria
9.0
10.0
8.0
6.5
Data Display Display for Kruskal-Wallis test. Gives sums of column ranks.
rs1
rs2
rs3
23.5000
21.0000
33.5000
Kruskal-Wallis Test: CCalls versus Soprano Test of equality of medians. High p-value means do
not reject equality of medians.
Kruskal-Wallis Test on CCalls
Ave
Soprano
N Median Rank
Z
Adelina
4
19.00
5.3 -0.85
Fifi
4
20.50
5.9 -0.42
Maria
4
29.00
8.4
1.27
Overall 12
6.5
H = 1.68 DF = 2 P = 0.431
H = 1.69 DF = 2 P = 0.430 (adjusted for ties)
* NOTE * One or more small samples
Data Display Display for Friedman test. Alternate columns give ranks within rows.
Row
1
2
3
4
Fifi
45
22
19
16
C31
2
2
1
1
Adelina
48
14
20
18
C32
3
1
2
2
Maria
30
32
28
22
C33
1
3
3
3
Data Display Display for Friedman test. Gives sums of ranks by column.
rs1
rs2
rs3
6.00000
8.00000
10.0000
79
252y0771t 11/28/07
Friedman Test: CCalls versus Soprano blocked by Venue Test of equality of medians blocked
by Venue. High p-value means do not reject equality of medians.
S = 2.00 DF = 2 P = 0.368
Sum of
Soprano N Est Median
Ranks
Adelina 4
22.000
8.0
Fifi
4
20.500
6.0
Maria
4
28.000
10.0
Grand median = 23.500
Data Display Display for Levene test. Original data.
Row
1
2
3
4
Fifi_1
45
22
19
16
Adelina_1
48
14
20
18
Maria_1
30
32
28
22
Data Display Display for Levene test. Column medians.
K41
K42
K43
20.5000
19.0000
29.0000
Data Display Display for Levene test. Columns less medians.
Row
1
2
3
4
Fifi_1
24.5
1.5
-1.5
-4.5
Adelina_1
29
-5
1
-1
Maria_1
1
3
-1
-7
Data Display Display for Levene test. Absolute value of Columns less medians.
Row
1
2
3
4
Fifi_1
24.5
1.5
1.5
4.5
Adelina_1
29
5
1
1
Maria_1
1
3
1
7
One-way ANOVA: Fifi_1, Adelina_1, Maria_1 Test of equality of variances. High p-value means do
not reject equality of variances.
Source DF
SS
MS
F
P
Factor
2
83
41 0.40 0.684
Error
9
937 104
Total
11 1020
S = 10.20
R-Sq = 8.11%
R-Sq(adj) = 0.00%
Individual 90% CIs For Mean Based on
Pooled StDev
Level
N Mean StDev ---------+---------+---------+---------+
Fifi_1
4 8.00 11.09
(------------*-------------)
Adelina_1 4 9.00 13.47
(-------------*------------)
Maria_1
4 3.00
2.83 (------------*-------------)
---------+---------+---------+---------+
0.0
7.0
14.0
21.0
Pooled StDev = 10.20
80
252y0771t 11/28/07
Test for Equal Variances: CCalls versus Soprano Test of equality of variances. High p-value
means do not reject equality of variances.
90% Bonferroni confidence intervals for standard deviations
Soprano N
Lower
StDev
Upper
Adelina 4 8.41036 15.5349 66.6522
Fifi 4 7.16184 13.2288 56.7577
Maria 4 2.33905
4.3205 18.5370
Bartlett's Test (Normal Distribution)
Test statistic = 3.53, p-value = 0.171
Levene's Test (Any Continuous Distribution)
Test statistic = 0.40, p-value = 0.684
Test for Equal Variances: CCalls versus Soprano
81