252meanx5 10/13/06
Examples of tests for differences of means
————— 10/13/2006 9:27:46 PM ————————————————————
Welcome to Minitab, press F1 for help.
MTB > WOpen "C:\Documents and Settings\RBOVE\My Documents\Minitab\252-D.MTW".
Retrieving worksheet from file: 'C:\Documents and Settings\RBOVE\My
Documents\Minitab\252-D.MTW'
Worksheet was saved on Fri Oct 13 2006
Results for: 252-D.MTW
MTB > Print c1 c2
Data Display
Row
1
2
3
4
5
6
7
8
9
10
11
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D1_1
305.47
306.48
324.22
334.75
320.61
299.40
299.40
293.34
322.63
294.35
319.60
287.28
290.31
302.43
263.04
307.48
323.64
304.45
310.51
312.53
288.29
307.48
291.32
319.60
328.69
284.25
310.51
314.55
300.41
309.50
303.44
332.73
329.70
321.62
293.34
324.65
322.63
317.58
311.52
311.52
337.78
357.98
291.32
272.13
326.67
305.46
The first data used for Method 1. This was done by method 3.
D1_2
344.53
344.53
344.53
341.47
305.77
298.63
292.51
305.77
389.41
322.09
328.21
361.87
319.03
337.39
291.49
350.65
341.47
369.01
348.61
338.41
309.85
330.25
370.03
327.19
298.63
367.99
293.53
338.41
332.29
341.47
338.41
334.33
329.23
321.07
336.37
348.61
318.01
310.87
319.03
301.69
342.49
344.53
340.45
307.81
333.31
326.17
1
252meanx5 10/13/06
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95
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98
99
100
101
102
103
104
105
106
107
108
109
277.18
304.45
304.45
320.61
313.54
295.36
315.56
282.23
304.45
270.11
301.42
308.49
331.72
293.34
252.94
316.57
340.81
297.38
307.48
304.45
290.31
304.45
293.34
285.26
303.44
333.74
323.64
280.21
288.29
270.11
264.05
302.43
293.34
274.15
272.13
334.75
277.18
318.59
284.25
303.44
303.44
273.14
273.14
286.27
352.93
274.15
286.27
311.52
292.33
262.03
284.25
311.52
276.17
286.27
331.72
311.52
331.72
290.31
288.29
317.58
315.56
320.61
297.38
303.73
354.73
342.49
310.87
323.11
339.43
350.65
353.71
303.73
320.05
329.23
315.97
332.29
356.77
312.91
325.15
310.87
324.13
324.13
329.23
348.61
320.05
342.49
346.57
369.01
330.25
305.77
364.93
341.47
321.07
327.19
333.31
331.27
316.99
314.95
336.37
303.73
345.55
318.01
322.09
331.27
336.37
332.29
324.13
318.01
323.11
329.23
327.19
335.35
303.73
313.93
369.01
335.35
319.03
311.89
310.87
346.57
348.61
322.09
334.33
347.59
338.41
343.51
2
252meanx5 10/13/06
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153
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169
282.23
318.59
292.33
286.27
299.40
283.24
277.18
310.51
284.25
305.46
291.32
293.34
304.45
298.39
289.30
276.17
305.46
295.36
285.26
289.30
299.40
299.40
313.54
294.35
272.13
320.61
301.42
311.52
293.34
328.69
255.97
282.23
285.26
280.21
291.32
297.38
268.09
295.36
313.54
275.16
288.29
305.46
281.22
259.00
301.42
291.32
317.58
280.21
284.25
354.95
273.14
319.60
286.27
269.10
292.33
286.27
277.18
313.54
292.33
324.65
300.67
326.17
335.35
311.89
309.85
347.59
372.07
347.59
341.47
334.33
308.83
313.93
316.99
331.27
330.25
339.43
385.33
307.81
303.73
325.15
290.47
314.95
311.89
316.99
349.63
312.91
334.33
352.69
324.13
321.07
338.41
333.31
327.19
327.19
333.31
3
252meanx5 10/13/06
MTB > describe c1 c2
Descriptive Statistics: D1_1, D1_2
Variable
D1_1
D1_2
N
169
144
N*
0
0
Variable
D1_1
D1_2
Maximum
357.98
389.41
Mean
300.00
330.00
SE Mean
1.54
1.58
StDev
20.00
18.98
Minimum
252.94
290.47
Q1
286.27
316.99
Median
299.40
329.74
Q3
313.54
341.47
MTB > TwoSample c1 c2.
Two-Sample T-Test and CI: D1_1, D1_2
Two-sample T for D1_1 vs D1_2
D1_1
D1_2
N
169
144
Mean
300.0
330.0
StDev
20.0
19.0
SE Mean
1.5
1.6
Difference = mu (D1_1) - mu (D1_2)
Estimate for difference: -30.0069
95% CI for difference: (-34.3481, -25.6657)
T-Test of difference = 0 (vs not =): T-Value = -13.60
307
P-Value = 0.000
DF =
MTB > print c3 c4
Data Display
Row
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
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20
21
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30
31
32
33
34
D1_3
15.752
19.680
20.662
17.716
21.644
15.752
16.734
20.662
16.734
17.716
19.680
24.590
21.644
17.716
13.788
16.734
17.716
21.644
15.752
24.590
17.716
14.770
11.824
17.716
24.590
23.608
18.698
18.698
18.698
19.680
21.644
13.788
14.770
23.608
The data in the second example for method 1.
D1_4
21.132
20.281
25.458
18.581
20.281
17.728
22.834
25.387
20.281
20.281
20.281
13.473
22.834
27.940
19.430
21.983
23.685
18.579
20.281
28.791
21.983
10.920
20.281
16.026
28.791
21.132
24.536
19.430
22.834
27.940
25.387
25.387
13.473
19.430
4
252meanx5 10/13/06
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93
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95
96
97
21.644
17.716
18.698
18.698
20.662
21.644
13.788
16.734
16.734
19.680
14.770
16.734
16.734
12.806
14.770
23.608
12.806
20.662
17.716
17.716
24.590
15.752
15.752
21.644
19.680
19.680
21.644
19.680
18.698
16.734
8.878
20.662
14.770
12.806
18.698
17.716
20.662
16.734
22.626
12.806
15.752
20.662
20.662
19.680
23.608
17.716
11.824
16.734
24.590
22.626
18.698
17.716
17.716
20.662
17.716
14.770
17.716
16.734
22.626
22.626
17.716
14.770
20.662
22.834
16.877
5
252meanx5 10/13/06
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108
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121
17.716
13.788
15.752
21.644
12.806
19.680
12.806
17.716
21.644
20.662
19.680
17.716
18.698
21.644
13.788
13.788
21.644
13.788
22.626
17.716
21.644
18.698
18.698
14.770
MTB > TwoSample c3 c4;
SUBC>
Alternative -1.
Two-Sample T-Test and CI: D1_3, D1_4
Two-sample T for D1_3 vs D1_4
D1_3
D1_4
N
121
36
Mean
18.40
21.30
StDev
3.30
4.20
SE Mean
0.30
0.70
Difference = mu (D1_3) - mu (D1_4)
Estimate for difference: -2.89917
95% upper bound for difference: -1.62189
T-Test of difference = 0 (vs <): T-Value = -3.81
P-Value = 0.000
DF = 48
MTB > print c5 c6
Data Display
Row
1
2
3
4
5
6
7
8
9
10
D2_1
53.32
58.44
49.32
53.24
25.60
61.34
41.56
60.49
44.47
21.19
The data used for method 2.
D2_2
45.02
74.04
116.14
103.76
67.90
85.44
60.88
67.90
71.41
63.52
6
252meanx5 10/13/06
MTB > TwoSample c5 c6;
SUBC>
Pooled;
SUBC>
Confidence 99.0.
Two-Sample T-Test and CI: D2_1, D2_2
The test is done with equal variances assumed
Two-sample T for D2_1 vs D2_2
D2_1
D2_2
N
10
10
Mean
46.9
75.6
StDev
14.0
21.0
SE Mean
4.4
6.6
Difference = mu (D2_1) - mu (D2_2)
Estimate for difference: -28.7040
99% CI for difference: (-51.6762, -5.7318)
T-Test of difference = 0 (vs not =): T-Value = -3.60
Both use Pooled StDev = 17.8456
P-Value = 0.002
DF = 18
MTB > TwoSample c5 c6;
SUBC>
Confidence 99.0.
Two-Sample T-Test and CI: D2_1, D2_2
The test is done with equal variances not assumed
Two-sample T for D2_1 vs D2_2
D2_1
D2_2
N
10
10
Mean
46.9
75.6
StDev
14.0
21.0
SE Mean
4.4
6.6
Difference = mu (D2_1) - mu (D2_2)
Estimate for difference: -28.7040
99% CI for difference: (-52.2211, -5.1869)
T-Test of difference = 0 (vs not =): T-Value = -3.60
P-Value = 0.003
DF = 15
MTB > print c7 c8
Data Display
Row
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
D3_1
5.67
8.08
8.31
7.63
7.01
8.24
4.84
7.14
10.41
9.16
6.98
10.72
11.88
5.82
8.27
11.04
The data used for method 3.
D3_2
10.37
7.16
3.65
6.98
6.84
4.51
4.55
9.82
7.21
8.64
6.72
9.10
9.08
4.16
7.87
6.94
MTB > describe c7 c8
Descriptive Statistics: D3_1, D3_2
Variable
D3_1
D3_2
N
16
16
N*
0
0
Variable
D3_1
D3_2
Maximum
11.880
10.370
Mean
8.200
7.100
SE Mean
0.506
0.512
StDev
2.025
2.049
Minimum
4.840
3.650
Q1
6.988
5.093
Median
8.160
7.070
Q3
10.098
8.970
7
252meanx5 10/13/06
MTB > TwoSample c7 c8.
Two-Sample T-Test and CI: D3_1, D3_2
Not assuming equal variances
Two-sample T for D3_1 vs D3_2
D3_1
D3_2
N
16
16
Mean
8.20
7.10
StDev
2.02
2.05
SE Mean
0.51
0.51
Difference = mu (D3_1) - mu (D3_2)
Estimate for difference: 1.10000
95% CI for difference: (-0.37310, 2.57310)
T-Test of difference = 0 (vs not =): T-Value = 1.53
P-Value = 0.138
DF = 29
MTB > TwoSample c7 c8;
SUBC>
Pooled.
Two-Sample T-Test and CI: D3_1, D3_2
Assuming equal variances
Two-sample T for D3_1 vs D3_2
D3_1
D3_2
N
16
16
Mean
8.20
7.10
StDev
2.02
2.05
SE Mean
0.51
0.51
Difference = mu (D3_1) - mu (D3_2)
Estimate for difference: 1.10000
95% CI for difference: (-0.37097, 2.57097)
T-Test of difference = 0 (vs not =): T-Value = 1.53
Both use Pooled StDev = 2.0372
P-Value = 0.137
DF = 30
MTB > print c9 - c11
Data Display
Row
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
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30
D4_1
4110.4
6692.4
13104.6
4378.3
7686.9
6221.8
5143.3
12332.8
9153.2
2403.2
9279.9
9283.9
23797.9
8036.2
10401.6
9996.9
5160.3
5849.7
11086.0
6484.2
11694.8
7558.8
11460.4
10342.2
6333.0
8465.1
11383.0
3304.7
2445.3
6558.6
One version of the data used for method 4.
D4_2
4117.4
6694.1
13110.3
4383.0
7689.2
6222.7
5142.2
12351.0
9156.0
2405.8
9274.3
9290.7
23806.3
8046.8
10397.1
9994.3
5170.3
5841.4
11086.9
6495.0
11692.4
7563.6
11469.7
10353.8
6337.2
8475.3
11388.1
3295.3
2455.0
6566.7
D4-D1
-7.00
-1.76
-5.73
-4.74
-2.35
-0.97
1.17
-18.15
-2.81
-2.60
5.61
-6.81
-8.35
-10.56
4.45
2.67
-10.03
8.25
-0.90
-10.84
2.38
-4.78
-9.28
-11.60
-4.15
-10.25
-5.07
9.48
-9.71
-8.12
8
252meanx5 10/13/06
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86
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90
91
92
93
9167.6
13339.6
9341.7
12766.1
17813.9
6532.8
14936.0
18102.2
8819.4
9293.3
5007.7
4834.8
7869.2
10013.0
6596.8
7629.4
11280.1
7191.2
7408.6
15711.9
12233.8
8955.3
2916.1
7354.9
16215.5
4381.1
10471.0
18366.7
8910.3
8872.6
7954.8
18437.2
5311.6
4688.1
3649.2
7488.0
7706.9
5817.1
14741.5
6199.4
6418.6
6458.2
23574.2
10627.7
8576.6
9501.9
8444.0
5861.1
19260.6
5048.6
9178.5
11334.1
7451.1
7935.2
3177.6
7526.2
3835.6
5054.1
14424.9
13181.5
8633.3
11906.2
11575.6
9172.5
13342.2
9343.0
12777.4
17816.5
6539.5
14934.8
18104.9
8820.3
9304.2
5014.9
4834.3
7869.6
10025.0
6605.3
7635.4
11281.0
7196.3
7416.5
15721.9
12237.1
8947.0
2921.6
7358.1
16223.3
4383.2
10483.0
18377.7
8911.8
8878.6
7963.7
18443.5
5319.1
4685.9
3659.3
7497.5
7716.2
5821.2
14745.8
6210.5
6430.9
6473.9
23570.1
10619.0
8576.6
9504.8
8452.0
5867.4
19268.0
5043.5
9180.1
11332.8
7452.0
7951.9
3176.7
7525.0
3842.2
5051.0
14419.6
13190.9
8633.8
11896.4
11577.4
-4.91
-2.56
-1.32
-11.32
-2.61
-6.77
1.28
-2.74
-0.84
-10.97
-7.21
0.49
-0.42
-11.98
-8.54
-6.01
-0.98
-5.14
-7.85
-10.01
-3.35
8.30
-5.55
-3.29
-7.74
-2.14
-12.01
-11.01
-1.41
-6.00
-8.87
-6.29
-7.47
2.11
-10.14
-9.41
-9.31
-4.09
-4.37
-11.03
-12.32
-15.71
4.10
8.65
0.02
-2.90
-8.05
-6.26
-7.39
5.10
-1.64
1.35
-0.84
-16.74
0.83
1.22
-6.62
3.10
5.39
-9.37
-0.54
9.75
-1.81
9
252meanx5 10/13/06
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95
96
97
98
99
100
8401.1
1371.3
7463.0
9644.3
10033.5
7506.8
9716.0
8399.6
1370.5
7466.1
9634.7
10041.7
7517.9
9718.9
1.55
0.80
-3.08
9.61
-8.21
-11.06
-2.94
MTB > Paired 'D4_1' 'D4_2';
SUBC>
Test 4.
Paired T-Test and CI: D4_1, D4_2
Paired T for D4_1 - D4_2
D4_1
D4_2
Difference
N
100
100
100
Mean
9075.97
9079.97
-4.00040
StDev
4315.72
4315.86
5.99961
SE Mean
431.57
431.59
0.59996
95% CI for mean difference: (-5.19085, -2.80995)
T-Test of mean difference = 4 (vs not = 4): T-Value = -13.33
P-Value = 0.000
MTB > print c12-c14
Data Display
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
29
30
31
32
33
34
35
36
37
38
39
D4_3
4110
6692
13105
4378
7687
6222
5143
12333
9153
2403
9280
9284
23798
8036
10402
9997
5160
5850
11086
6484
11695
7559
11460
10342
6333
8465
11383
3305
2445
6559
9168
13340
9342
12766
17814
6533
14936
18102
8819
D4_4
4117
6694
13110
4383
7689
6223
5142
12350
9156
2406
9274
9291
23806
8047
10397
9994
5169
5841
11088
6493
11692
7565
11472
10354
6338
8475
11388
3295
2455
6567
9173
13342
9343
12777
17816
6540
14935
18105
8820
A more realistic version of the Method 4 example
D4_D2
-7
-2
-5
-5
-2
-1
1
-17
-3
-3
6
-7
-8
-11
5
3
-9
9
-2
-9
3
-6
-12
-12
-5
-10
-5
10
-10
-8
-5
-2
-1
-11
-2
-7
1
-3
-1
10
252meanx5 10/13/06
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55
56
57
58
59
60
61
62
63
64
65
66
67
68
69
70
71
72
73
74
75
76
77
78
79
80
81
82
83
84
85
86
87
88
89
90
91
92
93
94
95
96
97
98
99
100
9293
5008
4835
7869
10013
6597
7629
11280
7191
7409
15712
12234
8955
2916
7355
16216
4381
10471
18367
8910
8873
7955
18437
5312
4688
3649
7488
7707
5817
14741
6199
6419
6458
23574
10628
8577
9502
8444
5861
19261
5049
9178
11334
7451
7935
3178
7526
3836
5054
14425
13182
8633
11906
11576
8401
1371
7463
9644
10033
7507
9716
9304
5015
4834
7870
10025
6605
7635
11281
7196
7416
15722
12237
8947
2922
7358
16223
4383
10483
18378
8912
8879
7964
18443
5319
4686
3659
7497
7716
5822
14746
6210
6431
6474
23570
10619
8578
9506
8452
5867
19268
5043
9180
11333
7452
7952
3177
7525
3842
5051
14420
13191
8634
11896
11577
8400
1370
7466
9635
10042
7518
9720
-11
-7
1
-1
-12
-8
-6
-1
-5
-7
-10
-3
8
-6
-3
-7
-2
-12
-11
-2
-6
-9
-6
-7
2
-10
-9
-9
-5
-5
-11
-12
-16
4
9
-1
-4
-8
-6
-7
6
-2
1
-1
-17
1
1
-6
3
5
-9
-1
10
-1
1
1
-3
9
-9
-11
-4
11
252meanx5 10/13/06
MTB > Paired 'D4_3' 'D4_4';
SUBC>
Test 4;
SUBC>
GHistogram;
SUBC>
GIndPlot;
SUBC>
GBoxplot.
Paired T-Test and CI: D4_3, D4_4
Paired T for D4_3 - D4_4
N
Mean
D4_3
100
9075.98
D4_4
100
9079.98
Difference 100 -4.00000
StDev
4315.75
4315.87
5.99495
SE Mean
431.57
431.59
0.59949
95% CI for mean difference: (-5.18953, -2.81047)
T-Test of mean difference = 4 (vs not = 4): T-Value = -13.34
Histogram of Differences
Individual Value Plot of Differences
Boxplot of Differences
MTB > NormTest 'D4_3';
SUBC>
Title "Before".
Before
MTB > NormTest 'D4_3';
SUBC>
Title "After".
P-Value = 0.000
This is shown below
Note that the graphs below show a definite
Skewness in both the before and after data
For the paired (Method 4) example. The graph
of the difference between them is somewhat
less skewed and gives a higher p-value for the
null hypothesis of Normality.
After
MTB > NormTest 'D4_D2';
SUBC>
Title "Difference".
Difference
12