Stat 3503/3602 — Unit 2: One-Factor ANOVA / Multiple Comparisons / Nonparametrics — Partial Solutions
2.1.1.
Enter the data into three appropriately labeled columns of a Minitab worksheet (unstacked format).
Print the data.
MTB > print c1 c2 c3
Data Display
Row
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
Beef
186
181
176
149
184
190
158
139
175
148
152
111
141
153
190
157
131
149
135
132
Meat
173
191
182
190
172
147
146
139
175
136
179
153
107
195
135
140
138
Poultry
129
132
102
106
94
102
87
99
170
113
135
142
86
143
152
146
144
2.1.2.
Just from looking at the number of digits in the observations, what do you suspect may be true of the
Poultry group?
Since there are several two digit numbers in the Poultry group and only three digit numbers in the Beef and Meat
groups, it appears that the mean caloric value of the Poultry hot dogs is likely lower than that of Beef or Meat.
2.2.1.
Use the menus to make a high resolution ("professional graphics") dotplots on the same scale. Discuss the
differences between this graphic and the collection of three dotplots shown above [in the questions].
Dotplot of Beef, Meat, Poultry
Beef
Meat
Poultry
96
112
128
144
Data
160
176
192
The differences between this compound dotplot and the one in the text are that the dots in this plot are larger and
easier to see, the scale is customized to the range of the data, and the font is more “professional.” (The
professional plot has a lot of wasted space, so it can accommodate more groups if necessary, and takes more
computer memory. When you cut/paste character graphs, include the line before and after the plot to avoid
"breaking" the graph, make sure the transferred plot has enough horizontal space, and appears in Courier type.)
Based on notes by Elizabeth Ellinger, Spring 2004, as expanded and modified by Bruce E. Trumbo, Winter 2005. Copyright © 2005 by Bruce E. Trumbo. All rights reserved.
Stat 3503/3602 — Unit 2: Partial Solutions
2
2.2.2.
Make a compact display of the data, involving numerical or graphical descriptive methods, suitable for
presentation in a report. The purpose is to display the important features of the data for a non-statistical audience.”
To communicate the crucial aspects of these data to a non-statistical audience, one might present the following:
Descriptive Statistics: Beef, Meat, Poultry
Variable
Beef
Meat
Poultry
Total
Count
20
17
17
Mean
156.85
158.71
122.47
Minimum
111.00
107.00
86.00
Maximum
190.00
195.00
170.00
Dotplot of Beef, Meat, Poultry
Beef
Meat
Poultry
96
112
128
144
Data
160
176
192
One might also include the standard deviation of each group. Dotplots contain more information than boxplots, but
for some purposes boxplots might be better. (But a really nonstatistical audience probably won't know what
standard deviations and quartiles are.)
Clearly, there is more than one right answer to this question. The point is to think about what descriptive methods
you are presenting and why.
2.3.1.
How would you cut and paste from your browser to enter the hot dog data into a single column? How would
you use the set command to enter the subscripts?”
To cut and paste from the browser into a single column, select a row of numbers and use ctl-C to copy them. Type
Set C4 in the MTB command window and paste the data after the DATA prompt. Click the enter key and repeat
until all the data has been entered. Type END when done.
To enter the subscripts, perform the following commands:
MTB > set c5
DATA> 20(1) 17(2) 17(3)
DATA> end
Based on notes by Elizabeth Ellinger, Spring 2004, as expanded and modified by Bruce E. Trumbo, Winter 2005. Copyright © 2005 by Bruce E. Trumbo. All rights reserved.
Stat 3503/3602 — Unit 2: Partial Solutions
3
2.3.2.
“Minitab boxplots for comparing groups can most conveniently be made using stacked data. Use the menus to
learn how to make three box plots on the same scale for the three groups of hot dog calorie measurements.”
To create the boxplots, select Graph ⇒ Boxplot ⇒ Multiple Y’s Simple and enter C1, C2, C3 in the graph variables
dialog box.
Boxplot of Beef, Meat, Poultry
200
175
Data
150
125
100
Beef
Meat
Poultry
Based on notes by Elizabeth Ellinger, Spring 2004, as expanded and modified by Bruce E. Trumbo, Winter 2005. Copyright © 2005 by Bruce E. Trumbo. All rights reserved.
Stat 3503/3602 — Unit 2: Partial Solutions
2.4.1.
4
Use the Fisher LSD method to interpret the pattern of differences among group means for the hot dog data.
The one way ANOVA with Fisher LSD multiple comparison:
One-way ANOVA: Hot Dogs versus Group
Source
Group
Error
Total
DF
2
51
53
S = 24.38
Level
1
2
3
N
20
17
17
SS
14491
30320
44811
MS
7245
595
F
12.19
R-Sq = 32.34%
Mean
156.85
158.71
122.47
P
0.000
R-Sq(adj) = 29.68%
StDev
22.64
25.24
25.48
Individual 95% CIs For Mean Based on
Pooled StDev
------+---------+---------+---------+--(-------*------)
(-------*-------)
(-------*-------)
------+---------+---------+---------+--120
135
150
165
Pooled StDev = 24.38
Fisher 95% Individual Confidence Intervals
All Pairwise Comparisons among Levels of Group
Simultaneous confidence level = 87.93%
Group = 1 subtracted from:
Group
2
3
Lower
-14.29
-50.53
Center
1.86
-34.38
Upper
18.00
-18.23
--------+---------+---------+---------+(-----*----)
(-----*----)
--------+---------+---------+---------+-30
0
30
60
Group = 2 subtracted from:
Group
3
Lower
-53.03
Center
-36.24
Upper
-19.45
--------+---------+---------+---------+(-----*-----)
--------+---------+---------+---------+-30
0
30
60
For both µ1 - µ3 and µ2 - µ3, zero is not in the confidence interval and, therefore, there is a significant difference
between groups 1 and 3 and groups 2 and 3. However, zero is contained in the confidence interval for µ1 - µ2 and,
therefore, there is no statistically significant difference between groups 1 and 2. These results are illustrated
graphically as:
Beef
Meat
Poultry
This is an "unbalanced" design: the groups are of unequal sizes. Thus the values of LSD may differ from one
comparison to another: Here the value of LSD used to compare Meat vs. Poultry will be different from the value
used to compare Meat vs. Beef.
1/2
1/2
LSD12 = t* sp (1/n1 + 1/n2) = (2.008)(24.38)(1/20 + 1/17) = 16.15. Compare with [18.00 – (–14.29)] / 2 = 16.15.
1/2
LSD23 = (2.008)(24.38)(1/17 + 1/17) = 16.79. Compare with [–19.45 – (–53.03)] / 2 = 16.79.
Based on notes by Elizabeth Ellinger, Spring 2004, as expanded and modified by Bruce E. Trumbo, Winter 2005. Copyright © 2005 by Bruce E. Trumbo. All rights reserved.
Stat 3503/3602 — Unit 2: Partial Solutions
2.4.3.
5
Use the Tukey HSD method to interpret the pattern of differences among group means.
The one way ANOVA results with the Tukey HSD test are as follows:
One-way ANOVA: Hot Dogs versus Group
Source
Group
Error
Total
DF
2
51
53
S = 24.38
Level
1
2
3
N
20
17
17
SS
14491
30320
44811
MS
7245
595
F
12.19
R-Sq = 32.34%
Mean
156.85
158.71
122.47
P
0.000
R-Sq(adj) = 29.68%
StDev
22.64
25.24
25.48
Individual 95% CIs For Mean Based on
Pooled StDev
------+---------+---------+---------+--(-------*------)
(-------*-------)
(-------*-------)
------+---------+---------+---------+--120
135
150
165
Pooled StDev = 24.38
Tukey 95% Simultaneous Confidence Intervals
All Pairwise Comparisons among Levels of Group
Individual confidence level = 98.05%
Group = 1 subtracted from:
Group
2
3
Lower
-17.54
-53.77
Center
1.86
-34.38
Upper
21.25
-14.98
---------+---------+---------+---------+
(------*-----)
(------*-----)
---------+---------+---------+---------+
-30
0
30
60
Group = 2 subtracted from:
Group
3
Lower
-56.40
Center
-36.24
Upper
-16.07
---------+---------+---------+---------+
(------*------)
---------+---------+---------+---------+
-30
0
30
60
The results for Tukey are the same as for Fisher with zero not in the confidence interval for µ1 - µ3 and µ2 - µ3 and
zero is in the confidence interval for µ1 - µ2. These results are illustrated graphically as:
Beef
Meat
Poultry
Strictly speaking, the Tukey procedure is meant only for balanced designs. See the approximation in Ott/
Longnecker for slightly unbalanced data.
Based on notes by Elizabeth Ellinger, Spring 2004, as expanded and modified by Bruce E. Trumbo, Winter 2005. Copyright © 2005 by Bruce E. Trumbo. All rights reserved.
Stat 3503/3602 — Unit 2: Partial Solutions
6
2.5.1.
Use menus to do Bartlett's test for homogeneity of variances. Say what menu path you used. What is your
conclusion? Why is doing Hartley's Fmax test by hand not an option here?
To test for equality of variances using the Bartlett’s test, the following menu commands are performed:
Stat ⇒ Basic Statistics ⇒ 2 Variances.
The output is as follows:
Test for Equal Variances for Hot Dogs
Bartlett's Test
Test Statistic
P-Value
1
0.29
0.863
Lev ene's Test
Group
Test Statistic
P-Value
0.49
0.613
2
3
15
20
25
30
35
40
95% Bonferroni Confidence Intervals for StDevs
45
The p-value for Bartlett's test is .863, which is greater than .05. Thus, the null hypothesis of equal variances cannot
be rejected.
To find out about Levene's test go to the ? on the menu bar and then do a search for "Levene". Here is part of the
explanation you will retrieve:
"The computational method for Levene's Test is a modification of Levene's procedure [2, 7]. This method
considers the distances of the observations from their sample median rather than their sample mean. Using
the sample median rather than the sample mean makes the test more robust for smaller samples."
Here "robust" means relatively unlikely to give an incorrect answer if the data are not normally distributed.
You can get explanations of most Minitab procedures in this way. Some of them (for example this one) are written
in a sufficiently user-friendly way as to be helpful, some are pretty obscure.
Based on notes by Elizabeth Ellinger, Spring 2004, as expanded and modified by Bruce E. Trumbo, Winter 2005. Copyright © 2005 by Bruce E. Trumbo. All rights reserved.
Stat 3503/3602 — Unit 2: Partial Solutions
7
2.5.2.
“Use menus (stacked one-way ANOVA) to find the "fits" for this ANOVA. How are the "fits" found for each
group in this case? Make a scatterplot of residuals vs. fits. (GRAPH ➯ Plot, Y=residuals, X=fits) and interpret the
result.”
The “fits” are the mean values for each group:
Beef:
156.850
Meat:
158.706
Poultry: 122.471
A scatterplot of residuals vs. fits follows:
Residuals Versus the Fitted Values
(response is Hot Dogs)
50
Residual
25
0
-25
-50
120
130
140
Fitted Value
150
160
If the residuals are a random sample from a normal population, their values should not be dependent upon the
mean of the group from which they are generated. From the above plot, it appears that the residual values are
independent of means.
Because the fits are distinct for each group (type of hot dog) the effect is something like a vertical dotplot of
residuals broken out by group.
Based on notes by Elizabeth Ellinger, Spring 2004, as expanded and modified by Bruce E. Trumbo, Winter 2005. Copyright © 2005 by Bruce E. Trumbo. All rights reserved.
Stat 3503/3602 — Unit 2: Partial Solutions
8
2.6.1.
An approximate nonparametric test results from ranking the data and performing a standard ANOVA on the
ranks (ignoring that the ranks are not normal). If we rank the data in 'Calories' then the smallest Calorie value will be
assigned rank 1 and the largest will be assigned rank 54. Below are commands to carry out the procedure, which gives
results similar to those already seen. Notice that the resulting confidence intervals are on the rank scale. Approximately
what are the Calorie values that correspond to the endpoints shown for the three confidence intervals?
MTB > name c15 'RnkCal'
MTB > rank 'Calorie' 'RnkCal'
MTB > onew 'RnkCal' 'Type'
The output of the ANOVA using rank as the response variable is as follows:
One-way ANOVA: RnkCal versus Group
Source
Group
Error
Total
DF
2
51
53
S = 13.41
Level
1
2
3
N
20
17
17
SS
3933
9177
13110
MS
1966
180
R-Sq = 30.00%
F
10.93
Mean
33.13
33.47
14.91
StDev
13.60
14.51
11.98
P
0.000
R-Sq(adj) = 27.25%
Individual 95% CIs For Mean Based on Pooled
StDev
+---------+---------+---------+--------(------*-------)
(-------*-------)
(--------*-------)
+---------+---------+---------+--------8.0
16.0
24.0
32.0
Pooled StDev = 13.41
1/2
Note that the above confidence intervals are calculated as y–i ± t.025,51sw/(ni) , where t.025,51 = 2.00758
and sW = 13.41. Intervals are of different lengths because the sample sizes differ. All three CIs are based
on the same variance estimate 13.41 – because we're assuming all three populations have the same variance.
To transform the above confidence intervals from the rank scale to the calorie scale, the approximate rank-valued
endpoint is replaced by the corresponding calorie value. This is common practice in communicating with clients,
who usually want to have information expressed in terms of their original measurements and may not even know
what ranks are.
Grou
p
1
2
3
Left Rank
Endpoint
27.11
26.94
8.38
Right Rank
Endpoint
39.15
40.00
21.44
Left Calorie
Endpoint
145
145
108
Right Calorie
Endpoint
170
172
139
These can be obtained from the sorted list observations with their respective ranks shown below:
Based on notes by Elizabeth Ellinger, Spring 2004, as expanded and modified by Bruce E. Trumbo, Winter 2005. Copyright © 2005 by Bruce E. Trumbo. All rights reserved.
Stat 3503/3602 — Unit 2: Partial Solutions
RnkCal
1.0
2.0
3.0
4.0
5.5
5.5
7.0
8.0
9.0
10.0
11.0
12.0
13.5
13.5
16.0
16.0
16.0
18.0
19.0
20.5
20.5
22.0
23.0
24.0
25.0
26.0
27.5
27.5
Calories
86
87
94
99
102
102
106
107
111
113
129
131
132
132
135
135
135
136
138
139
139
140
141
142
143
144
146
146
RnkCal
29.0
30.0
31.5
31.5
33.5
33.5
35.5
35.5
37.0
38.0
39.0
40.0
41.0
42.5
42.5
44.0
45.0
46.0
47.0
48.0
49.0
51.0
51.0
51.0
53.0
54.0
9
Calories
147
148
149
149
152
152
153
153
157
158
170
172
173
175
175
176
179
181
182
184
186
190
190
190
191
195
Based on notes by Elizabeth Ellinger, Spring 2004, as expanded and modified by Bruce E. Trumbo, Winter 2005. Copyright © 2005 by Bruce E. Trumbo. All rights reserved.
Stat 3503/3602 — Unit 2: Partial Solutions
10
2.7.1.
Use Minitab's capability to generate random samples. Sample 10 observations from each of 15 groups, all
known to have a normal distribution with mean 100 and standard deviation 10. Thus, we know that there are no real
differences among means. Yet, by comparing the two groups that happen to have the largest and smallest group sample
means, we can often find a bogus significant difference with the Fisher procedure. In practice, you shouldn't look at
Fisher LSDs unless the main F-test rejects.
Here is a (slightly simplified, but still accurate) version of the commands generated by the menu steps in the
questions. The advantage of using commands is that, once entered and used, the commands can be cut from the
Worksheet and pasted at the active MTB > prompt (at the very bottom of the material in Session window) and used
again for another simulation without any further typing. The rest of the prompts will appear when you press Enter.
MTB >
DATA>
DATA>
MTB >
SUBC>
MTB >
SUBC>
set c22
1(1:15)10
end.
rand 150 c21;
norm 100 10.
onew C21 C22;
fisher.
Here is printout for one run. You are looking for Fisher CIs that don't cover 0. (An early hint of where to look is from
the default CIs at the start; look among intervals that don't cover means of other intervals.)
There is a lot of output from each run. Here we highlight the Fisher CIs that don't cover 0. There is no guarantee
that your first run will produce any such Fisher CIs, but there is a fairly high probability. If you don't get any on the
first run, try again. We show two runs: The first has an example (close call), the second has stronger examples.
First run. All output shown.
One-way ANOVA: C21 versus C22
Source
C22
Error
Total
DF
14
135
149
S = 9.315
Level
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
N
10
10
10
10
10
10
10
10
10
10
10
10
10
10
10
SS
1103.7
11712.8
12816.5
MS
78.8
86.8
R-Sq = 8.61%
Mean
101.31
98.78
104.10
99.47
98.12
99.41
97.58
102.35
96.24
99.26
104.07
104.27
101.13
104.52
104.29
StDev
9.76
9.32
9.45
6.21
6.84
7.18
9.88
10.12
7.56
10.12
11.91
10.31
7.99
12.41
8.20
F
0.91
P
0.551
Note: Not Significant! Shouldn't even look at Fisher LSD.
R-Sq(adj) = 0.00%
Individual 95% CIs For Mean Based on
Pooled StDev
---------+---------+---------+---------+
(-----------*----------)
(-----------*----------)
(----------*-----------)
(-----------*-----------)
(----------*-----------)
(-----------*----------)
(----------*-----------)
(-----------*----------)
(----------*-----------)
(-----------*----------)
(-----------*-----------)
(-----------*----------)
(----------*-----------)
(-----------*-----------)
(-----------*----------)
---------+---------+---------+---------+
95.0
100.0
105.0
110.0
Pooled StDev = 9.31
Based on notes by Elizabeth Ellinger, Spring 2004, as expanded and modified by Bruce E. Trumbo, Winter 2005. Copyright © 2005 by Bruce E. Trumbo. All rights reserved.
Stat 3503/3602 — Unit 2: Partial Solutions
11
Fisher 95% Individual Confidence Intervals
All Pairwise Comparisons among Levels of C22
Simultaneous confidence level = 19.24%
C22 =
C22
2
3
4
5
6
7
8
9
10
11
12
13
14
15
1 subtracted from:
C22 =
Lower
-10.770
-5.449
-10.082
-11.433
-10.147
-11.976
-7.202
-13.311
-10.289
-5.479
-5.280
-8.418
-5.036
-5.267
Upper
5.706
11.028
6.394
5.043
6.330
4.500
9.275
3.166
6.188
10.997
11.197
8.058
11.440
11.209
-------+---------+---------+---------+-(-------*--------)
(-------*-------)
(-------*-------)
(-------*-------)
(-------*-------)
(-------*--------)
(-------*-------)
(-------*-------)
(-------*-------)
(-------*-------)
(-------*-------)
(-------*-------)
(-------*-------)
(-------*-------)
-------+---------+---------+---------+--10
0
10
20
2 subtracted from:
C22
3
4
5
6
7
8
9
10
11
12
13
14
15
Lower
-2.916
-7.550
-8.901
-7.614
-9.444
-4.669
-10.779
-7.756
-2.947
-2.747
-5.886
-2.504
-2.735
C22 =
C22
4
5
6
7
8
9
10
11
12
13
14
15
Center
-2.532
2.790
-1.844
-3.195
-1.908
-3.738
1.037
-5.073
-2.050
2.759
2.959
-0.180
3.202
2.971
Center
5.322
0.688
-0.663
0.624
-1.206
3.569
-2.540
0.482
5.291
5.491
2.352
5.734
5.503
Upper
13.560
8.926
7.575
8.862
7.032
11.807
5.698
8.720
13.530
13.729
10.590
13.972
13.741
-------+---------+---------+---------+-(-------*--------)
(--------*-------)
(-------*--------)
(--------*-------)
(-------*-------)
(--------*-------)
(-------*--------)
(-------*--------)
(-------*--------)
(-------*--------)
(-------*--------)
(--------*-------)
(--------*-------)
-------+---------+---------+---------+--10
0
10
20
3 subtracted from:
Lower
-12.872
-14.223
-12.936
-14.766
-9.991
-16.100
-13.078
-8.269
-8.069
-11.208
-7.826
-8.057
Center
-4.634
-5.985
-4.698
-6.528
-1.753
-7.862
-4.840
-0.031
0.169
-2.970
0.412
0.181
Upper
3.605
2.254
3.540
1.711
6.485
0.376
3.398
8.208
8.407
5.269
8.651
8.420
-------+---------+---------+---------+-(-------*--------)
(-------*-------)
(-------*--------)
(-------*--------)
(-------*-------)
(-------*-------)
(Close: RH end barely positive.)
(-------*-------)
(-------*-------)
(-------*-------)
(-------*-------)
(-------*--------)
(-------*-------)
-------+---------+---------+---------+--10
0
10
20
Based on notes by Elizabeth Ellinger, Spring 2004, as expanded and modified by Bruce E. Trumbo, Winter 2005. Copyright © 2005 by Bruce E. Trumbo. All rights reserved.
Stat 3503/3602 — Unit 2: Partial Solutions
C22 =
4 subtracted from:
C22
5
6
7
8
9
10
11
12
13
14
15
Lower
-9.589
-8.302
-10.132
-5.357
-11.467
-8.444
-3.635
-3.435
-6.574
-3.192
-3.423
C22 =
Center
-1.351
-0.064
-1.894
2.881
-3.228
-0.206
4.603
4.803
1.664
5.046
4.815
Upper
6.887
8.174
6.344
11.119
5.010
8.032
12.842
13.041
9.902
13.284
13.053
-------+---------+---------+---------+-(--------*-------)
(-------*-------)
(-------*-------)
(-------*-------)
(-------*-------)
(-------*-------)
(--------*-------)
(-------*-------)
(--------*-------)
(-------*-------)
(-------*-------)
-------+---------+---------+---------+--10
0
10
20
5 subtracted from:
C22
6
7
8
9
10
11
12
13
14
15
Lower
-6.952
-8.781
-4.007
-10.116
-7.094
-2.284
-2.084
-5.223
-1.841
-2.072
C22 =
Center
1.287
-0.543
4.232
-1.878
1.145
5.954
6.154
3.015
6.397
6.166
Upper
9.525
7.695
12.470
6.361
9.383
14.192
14.392
11.253
14.635
14.404
-------+---------+---------+---------+-(-------*--------)
(-------*--------)
(-------*-------)
(-------*-------)
(-------*-------)
(-------*-------)
(-------*-------)
(-------*-------)
(-------*--------)
(-------*-------)
-------+---------+---------+---------+--10
0
10
20
6 subtracted from:
C22
7
8
9
10
11
12
13
14
15
Lower
-10.068
-5.293
-11.403
-8.380
-3.571
-3.371
-6.510
-3.128
-3.359
C22 =
C22
8
9
10
11
12
13
14
15
12
Center
-1.830
2.945
-3.164
-0.142
4.667
4.867
1.728
5.110
4.879
Upper
6.409
11.183
5.074
8.096
12.906
13.105
9.967
13.349
13.118
-------+---------+---------+---------+-(-------*-------)
(-------*-------)
(-------*-------)
(-------*-------)
(--------*-------)
(-------*-------)
(--------*-------)
(-------*-------)
(-------*-------)
-------+---------+---------+---------+--10
0
10
20
7 subtracted from:
Lower
-3.463
-9.573
-6.551
-1.741
-1.541
-4.680
-1.298
-1.529
Center
4.775
-1.334
1.688
6.497
6.697
3.558
6.940
6.709
Upper
13.013
6.904
9.926
14.736
14.935
11.796
15.178
14.947
-------+---------+---------+---------+-(-------*-------)
(--------*-------)
(--------*-------)
(-------*--------)
(--------*-------)
(--------*-------)
(-------*-------)
(--------*-------)
-------+---------+---------+---------+--10
0
10
20
Based on notes by Elizabeth Ellinger, Spring 2004, as expanded and modified by Bruce E. Trumbo, Winter 2005. Copyright © 2005 by Bruce E. Trumbo. All rights reserved.
Stat 3503/3602 — Unit 2: Partial Solutions
C22 =
8 subtracted from:
C22
9
10
11
12
13
14
15
Lower
-14.348
-11.325
-6.516
-6.316
-9.455
-6.073
-6.304
C22 =
C22
10
11
12
13
14
15
13
Center
-6.109
-3.087
1.722
1.922
-1.217
2.165
1.934
Upper
2.129
5.151
9.961
10.160
7.022
10.404
10.172
-------+---------+---------+---------+-(-------*-------)
(-------*-------)
(--------*-------)
(-------*-------)
(-------*-------)
(-------*-------)
(-------*-------)
-------+---------+---------+---------+--10
0
10
20
9 subtracted from:
Lower
-5.216
-0.407
-0.207
-3.346
0.036
-0.195
Center
3.022
7.832
8.031
4.893
8.275
8.043
Upper
11.261
16.070
16.270
13.131
16.513
16.282
-------+---------+---------+---------+-(-------*-------)
(-------*-------)
(Close: LH end barely negative)
(-------*-------)
(Close: LH end barely negative)
(-------*-------)
(-------*--------)
Borderline example: Look at numbers
(-------*-------)
(LH end barely negative)
-------+---------+---------+---------+--10
0
10
20
C22 = 10 subtracted from:
C22
11
12
13
14
15
Lower
-3.429
-3.229
-6.368
-2.986
-3.217
Center
4.809
5.009
1.870
5.252
5.021
Upper
13.048
13.247
10.109
13.491
13.260
-------+---------+---------+---------+-(-------*-------)
(-------*-------)
(-------*-------)
(-------*-------)
(-------*-------)
-------+---------+---------+---------+--10
0
10
20
C22 = 11 subtracted from:
C22
12
13
14
15
Lower
-8.039
-11.177
-7.795
-8.027
Center
0.200
-2.939
0.443
0.212
Upper
8.438
5.299
8.681
8.450
-------+---------+---------+---------+-(-------*-------)
(-------*-------)
(-------*--------)
(-------*-------)
-------+---------+---------+---------+--10
0
10
20
C22 = 12 subtracted from:
C22
13
14
15
Lower
-11.377
-7.995
-8.226
Center
-3.139
0.243
0.012
Upper
5.100
8.482
8.250
-------+---------+---------+---------+-(-------*-------)
(-------*-------)
(-------*-------)
-------+---------+---------+---------+--10
0
10
20
C22 = 13 subtracted from:
C22
14
15
Lower
-4.856
-5.087
Center
3.382
3.151
Upper
11.620
11.389
-------+---------+---------+---------+-(-------*--------)
(-------*-------)
-------+---------+---------+---------+--10
0
10
20
Based on notes by Elizabeth Ellinger, Spring 2004, as expanded and modified by Bruce E. Trumbo, Winter 2005. Copyright © 2005 by Bruce E. Trumbo. All rights reserved.
Stat 3503/3602 — Unit 2: Partial Solutions
14
C22 = 14 subtracted from:
C22
15
Lower
-8.469
Center
-0.231
Upper
8.007
-------+---------+---------+---------+-(-------*-------)
-------+---------+---------+---------+--10
0
10
20
Second run. Here we save space by showing only the interesting clumps of output.
MTB >
DATA>
DATA>
MTB >
SUBC>
MTB >
SUBC>
set c22
1(1:15)10
end.
rand 150 c21;
norm 100 10.
onew C21 C22;
fisher.
One-way ANOVA: C21 versus C22
Source
C22
Error
Total
DF
14
135
149
S = 10.42
Level
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
N
10
10
10
10
10
10
10
10
10
10
10
10
10
10
10
SS
1748
14653
16401
MS
125
109
R-Sq = 10.66%
F
1.15
Mean
99.96
99.41
100.00
96.66
101.82
95.18
103.64
97.17
103.94
100.03
98.30
99.35
98.53
101.46
89.65
StDev
9.29
14.84
13.15
10.06
10.18
10.41
12.28
7.79
8.42
9.68
8.49
8.98
7.73
12.66
9.32
P
0.321
Main F-test not significant.
R-Sq(adj) = 1.39%
Individual 95% CIs For Mean Based on
Pooled StDev
-+---------+---------+---------+-------(---------*--------)
(--------*--------)
(--------*--------)
(--------*--------)
(--------*---------)
(--------*--------)
(--------*--------)
(--------*--------)
(--------*---------)
(--------*--------)
(--------*---------)
(--------*--------)
(---------*--------)
(--------*--------)
(--------*--------)
(Involved in all 3 examples)
-+---------+---------+---------+-------84.0
91.0
98.0
105.0
Pooled StDev = 10.42
Based on notes by Elizabeth Ellinger, Spring 2004, as expanded and modified by Bruce E. Trumbo, Winter 2005. Copyright © 2005 by Bruce E. Trumbo. All rights reserved.
Stat 3503/3602 — Unit 2: Partial Solutions
15
Fisher 95% Individual Confidence Intervals
All Pairwise Comparisons among Levels of C22
Simultaneous confidence level = 19.24%
...
C22 =
C22
8
9
10
11
12
13
14
15
7 subtracted from:
Lower
-15.68
-8.91
-12.82
-14.55
-13.50
-14.32
-11.39
-23.20
Center
-6.46
0.30
-3.60
-5.34
-4.28
-5.11
-2.17
-13.99
Upper
2.75
9.51
5.61
3.88
4.93
4.11
7.04
-4.77
+---------+---------+---------+--------(-------*------)
(------*-------)
(-------*-------)
(-------*------)
(------*-------)
(-------*------)
(------*-------)
(------*-------)
+---------+---------+---------+---------24
-12
0
12
Strong example
...
C22 =
C22
10
11
12
13
14
15
9 subtracted from:
Lower
-13.12
-14.85
-13.80
-14.62
-11.69
-23.50
Center
-3.90
-5.64
-4.58
-5.41
-2.47
-14.29
Upper
5.31
3.58
4.63
3.81
6.74
-5.07
+---------+---------+---------+--------(-------*------)
(------*-------)
(------*-------)
(------*-------)
(-------*-------)
(-------*-------)
+---------+---------+---------+---------24
-12
0
12
Strong example
...
C22 = 14 subtracted from:
C22
15
Lower
-21.03
Center
-11.81
Upper
-2.60
+---------+---------+---------+--------(-------*-------)
+---------+---------+---------+---------24
-12
0
12
Strong example
Note: Working at the 5% level you would expect that once in 20 runs the main F-test would give a P-value less than
5%, leading to a totally wrong interpretation (5% = 1/20).
Based on notes by Elizabeth Ellinger, Spring 2004, as expanded and modified by Bruce E. Trumbo, Winter 2005. Copyright © 2005 by Bruce E. Trumbo. All rights reserved.