252y0611s1 10/3/06 (Open in ‘Print Layout’ format)
Personalized Computations for Problem in Section II and First Problem
in Section IV
Computations for Mean – Variance Problem
II. (5 points-2 point penalty for not trying part a.) In order not to violate the truth in labeling law a teabag
must contain at least 5.5 oz of tea. A sample of 9 items is taken from a large number of tea bags. The data
below is found. (Recomputing what I’ve done for you is a great way to waste time.) b) and d) require
statistical tests.
a. Compute the sample standard deviation, s , of the waiting times. Show your work! (2)
b. Is the population mean significantly below 5.5 (Use a 95% confidence level)? Show your
work! (3)
c. (Extra Credit) Find an approximate p value for your null hypothesis. (2)
i. Assume that the Normal distribution does not apply and, using your data, test that the median is
above 128. (3)
[12]
j. (Extra credit) Use your data to find an approximate 90% 2-sided confidence interval for the
median.
————— 10/3/2006 10:14:03 PM ————————————————————
Welcome to Minitab, press F1 for help.
MTB > WOpen "C:\Documents and Settings\RBOVE\My Documents\Minitab\Teabags.mtw".
Retrieving worksheet from file: 'C:\Documents and Settings\RBOVE\My
Documents\Minitab\Teabags.mtw'
Worksheet was saved on Wed May 12 1999
Results for: 252x061-2.mtb
(t1 is on version 1, t2 on version 2)
MTB > Onet c2;
SUBC> Test 5.5;
SUBC> Alternative -1.
One-Sample T: t1
Test of mu = 5.5 vs < 5.5
Variable
t1
N
9
Mean
5.40667
StDev
0.12217
SE Mean
0.04072
95%
Upper
Bound
5.48239
T
-2.29
P
0.026
SE Mean
0.03670
95%
Upper
Bound
5.50825
T
-1.63
P
0.070
MTB > Onet c3;
SUBC> Test 5.5;
SUBC> Alternative -1.
One-Sample T: t2
Test of mu = 5.5 vs < 5.5
Variable
t2
N
9
Mean
5.44000
MTB > let c4 = c2*c2
MTB > let c5 = c3*c3
MTB > print c2 c4
Data Display
Row
1
2
3
4
t1
5.36
5.17
5.40
5.38
t1sq
28.7296
26.7289
29.1600
28.9444
StDev
0.11011
252y0611s1 10/3/06 (Open in ‘Print Layout’ format)
5
6
7
8
9
5.46
5.50
5.62
5.35
5.42
29.8116
30.2500
31.5844
28.6225
29.3764
MTB > sum c2
Sum of t1
Sum of t1 = 48.66
MTB > sum c4
Sum of t1sq
Sum of t1sq = 263.208
#263.2078
MTB > let c6 = 10000 * c4
MTB > sum c6
Sum of t1sqm
Sum of t1sqm = 2632078
MTB > print c3 c5
Data Display
Row
1
2
3
4
5
6
7
8
9
t2
5.35
5.68
5.35
5.47
5.31
5.49
5.41
5.48
5.42
t2sq
28.6225
32.2624
28.6225
29.9209
28.1961
30.1401
29.2681
30.0304
29.3764
MTB > sum c3
Sum of t2
Sum of t2 = 48.96
MTB > sum c5
Sum of t2sq
Sum of t2sq = 266.439
MTB > let c7 = 10000 * c5
MTB > sum c7
Sum of t2sqm
Sum of t2sqm = 2664394
#266.4394
252y0611s1 10/3/06 (Open in ‘Print Layout’ format)
Computations for First Problem in Section IV
1.
(Moore, Notz) You are thinking that it may be desirable to start a wellness program for your
(large) company. You are told that the company will only start such a program if you can show
that the blood pressure of a group of mid-level executives is above normal. The individuals are all
between 35 and 44 years old and US statistics show that mean systolic blood pressure for men in
that age range is 128. You take a sample of 72 executives and get the following results.
x
139.60
136.21
114.18
128.45
120.68
127.51
128.07
161.43
161.09
130.61
130.15
154.74
126.29
124.48
117.51
138.22
169.05
116.14
182.53
141.96
122.14
163.18
124.61
100.03
130.77
140.35
158.74
120.02
137.23
127.22
141.54
105.67
149.55
109.52
131.40
126.54
118.77
141.15
150.30
126.93
144.71
127.32
136.69
125.06
135.21
149.44
133.89
118.37
124.80
133.00
131.74
135.69
169.61
126.71
107.30
122.73
125.35
152.64
109.62
116.59
132.00
117.84
120.01
117.47
145.25
159.94
112.34
145.10
119.39
127.67
117.97
112.40
To personalize the data below take the last digit of your student number, divide it by 10 and add it to the
numbers below. If the last digit of your student number is zero, add 1.00. Label the problem ‘Version 1,’
‘Version 2,’ … ‘Version 10’ according to the number that you used. (For example, Seymour Butz’s
student number is 976502, so he will add 0.20 and change the data to 139.84, 130.81, 182.73 etc. – but see
the hint below, you do not need to write down all the numbers that you are using, just your computations.)
x  9528 .41
Hint - if you use the computational formula: For the original numbers n  72 ,
and
x

2
 1280763 .7 . If you add a quantity a to a column of numbers,
 x  a   x na,  x  a    x  2a x na
2
2
2
Assume that the Normal distribution applies to the data and use a 99% confidence level.
a. Find the sample mean and sample standard deviation of the incomes in your data, showing
your work. (1) (Your mean should be fairly near 132 and your sample standard deviation should
be near 16 or 17.)
b. State your null and alternative hypotheses (1)
c. Test the hypothesis using a test ratio (1)
d. Test the hypothesis using a critical value for a sample mean. (1)
e. Test the hypothesis using a confidence interval (1)
f. Find an approximate p-value for the null hypothesis. (1)
g. On the basis of your tests, will you get a wellness program? Why? (1)
————— 10/3/2006 4:45:21 PM ————————————————————
Welcome to Minitab, press F1 for help.
MTB > WOpen "C:\Documents and Settings\RBOVE\My Documents\Minitab\252x06101.MTW".
Retrieving worksheet from file: 'C:\Documents and Settings\RBOVE\My
Documents\Minitab\252x061-01.MTW'
Worksheet was saved on Tue Oct 03 2006
Results for: 252x061-01.MTW
MTB > let c12 = c1 * c1
MTB > sum c1
Sum of s0
Sum of s0 = 9528.41
x02 is in column c12. Since all variances are identical, I only worked on
doing this one.
252y0611s1 10/3/06 (Open in ‘Print Layout’ format)
MTB > ssq c1
Sum of Squares of s0
Sum of squares (uncorrected) of s0 = 1280764
MTB > sum c12
Sum of s0sq
Sum of s0sq = 1280764
MTB > let c13 = c12 * 10
MTB > sum c13
This was done to find the next digit in the sum of squares.
Sum of C13
Sum of C13 = 12807637
MTB > ssq c2
Sum of Squares of s1
s1-s11 are the columns for Versions 1-10.
Sum of squares (uncorrected) of s1 = 1282670
MTB > ssq c3
Sum of Squares of s2
Sum of squares (uncorrected) of s2 = 1284578
MTB > ssq c4
Sum of Squares of s3
Sum of squares (uncorrected) of s3 = 1286487
MTB > ssq c5
Sum of Squares of s4
Sum of squares (uncorrected) of s4 = 1288398
MTB > ssq c6
Sum of Squares of s5
Sum of squares (uncorrected) of s5 = 1290310
MTB > ssq c7
Sum of Squares of s6
Sum of squares (uncorrected) of s6 = 1292224
MTB > ssq c8
Sum of Squares of s7
Sum of squares (uncorrected) of s7 = 1294139
MTB > ssq c9
Sum of Squares of s8
Sum of squares (uncorrected) of s8 = 1296055
MTB > ssq c10
Sum of Squares of s9
Sum of squares (uncorrected) of s9 = 1297973
MTB > ssq c11
Sum of Squares of s10
Sum of squares (uncorrected) of s10 = 1299893
MTB > sum c2
Sum of s1
Sum of s1 = 9535.61
252y0611s1 10/3/06 (Open in ‘Print Layout’ format)
MTB > sum c3
Sum of s2
Sum of s2 = 9542.81
MTB > sum c4
Sum of s3
Sum of s3 = 9550.01
MTB > sum c5
Sum of s4
Sum of s4 = 9557.21
MTB > sum c6
Sum of s5
Sum of s5 = 9564.41
MTB > sum c7
Sum of s6
Sum of s6 = 9571.61
MTB > sum c8
Sum of s7
Sum of s7 = 9578.81
MTB > sum c9
Sum of s8
Sum of s8 = 9586.01
MTB > sum c10
Sum of s9
Sum of s9 = 9593.21
MTB > sum c11
Sum of s10
Sum of s10 = 9600.41
MTB > print c1 c12
Data Display
Row
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
s0
139.60
130.61
182.53
120.02
118.77
149.44
107.30
117.47
136.21
130.15
141.96
137.23
141.15
133.89
122.73
145.25
114.18
154.74
122.14
127.22
s0sq
19488.2
17059.0
33317.2
14404.8
14106.3
22332.3
11513.3
13799.2
18553.2
16939.0
20152.6
18832.1
19923.3
17926.5
15062.7
21097.6
13037.1
23944.5
14918.2
16184.9
The data for version 0.
252y0611s1 10/3/06 (Open in ‘Print Layout’ format)
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
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
150.30
118.37
125.35
159.94
128.45
126.29
163.18
141.54
126.93
124.80
152.64
112.34
120.68
124.48
124.61
105.67
144.71
133.00
109.62
145.10
127.51
117.51
100.03
149.55
127.32
131.74
116.59
119.39
128.07
138.22
130.77
109.52
136.69
135.69
132.00
127.67
161.43
169.05
140.35
131.40
125.06
169.61
117.84
117.97
161.09
116.14
158.74
126.54
135.21
126.71
120.01
112.40
22590.1
14011.5
15712.6
25580.8
16499.4
15949.2
26627.7
20033.6
16111.2
15575.0
23299.0
12620.3
14563.7
15495.3
15527.7
11166.1
20941.0
17689.0
12016.5
21054.0
16258.8
13808.6
10006.0
22365.2
16210.4
17355.4
13593.2
14254.0
16401.9
19104.8
17100.8
11994.6
18684.2
18411.8
17424.0
16299.6
26059.6
28577.9
19698.1
17266.0
15640.0
28767.6
13886.3
13916.9
25950.0
13488.5
25198.4
16012.4
18281.7
16055.4
14402.4
12633.8
MTB > print c2 - c12
Data Display
Row
1
2
3
4
5
6
7
s1
139.70
130.71
182.63
120.12
118.87
149.54
107.40
The data for versions 1-10
s2
139.80
130.81
182.73
120.22
118.97
149.64
107.50
s3
139.90
130.91
182.83
120.32
119.07
149.74
107.60
s4
140.00
131.01
182.93
120.42
119.17
149.84
107.70
s5
140.10
131.11
183.03
120.52
119.27
149.94
107.80
s6
140.20
131.21
183.13
120.62
119.37
150.04
107.90
s7
140.30
131.31
183.23
120.72
119.47
150.14
108.00
s8
140.40
131.41
183.33
120.82
119.57
150.24
108.10
s9
140.50
131.51
183.43
120.92
119.67
150.34
108.20
252y0611s1 10/3/06 (Open in ‘Print Layout’ format)
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
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
117.57
136.31
130.25
142.06
137.33
141.25
133.99
122.83
145.35
114.28
154.84
122.24
127.32
150.40
118.47
125.45
160.04
128.55
126.39
163.28
141.64
127.03
124.90
152.74
112.44
120.78
124.58
124.71
105.77
144.81
133.10
109.72
145.20
127.61
117.61
100.13
149.65
127.42
131.84
116.69
119.49
128.17
138.32
130.87
109.62
136.79
135.79
132.10
127.77
161.53
169.15
140.45
131.50
125.16
169.71
117.94
118.07
161.19
116.24
158.84
126.64
135.31
126.81
117.67
136.41
130.35
142.16
137.43
141.35
134.09
122.93
145.45
114.38
154.94
122.34
127.42
150.50
118.57
125.55
160.14
128.65
126.49
163.38
141.74
127.13
125.00
152.84
112.54
120.88
124.68
124.81
105.87
144.91
133.20
109.82
145.30
127.71
117.71
100.23
149.75
127.52
131.94
116.79
119.59
128.27
138.42
130.97
109.72
136.89
135.89
132.20
127.87
161.63
169.25
140.55
131.60
125.26
169.81
118.04
118.17
161.29
116.34
158.94
126.74
135.41
126.91
117.77
136.51
130.45
142.26
137.53
141.45
134.19
123.03
145.55
114.48
155.04
122.44
127.52
150.60
118.67
125.65
160.24
128.75
126.59
163.48
141.84
127.23
125.10
152.94
112.64
120.98
124.78
124.91
105.97
145.01
133.30
109.92
145.40
127.81
117.81
100.33
149.85
127.62
132.04
116.89
119.69
128.37
138.52
131.07
109.82
136.99
135.99
132.30
127.97
161.73
169.35
140.65
131.70
125.36
169.91
118.14
118.27
161.39
116.44
159.04
126.84
135.51
127.01
117.87
136.61
130.55
142.36
137.63
141.55
134.29
123.13
145.65
114.58
155.14
122.54
127.62
150.70
118.77
125.75
160.34
128.85
126.69
163.58
141.94
127.33
125.20
153.04
112.74
121.08
124.88
125.01
106.07
145.11
133.40
110.02
145.50
127.91
117.91
100.43
149.95
127.72
132.14
116.99
119.79
128.47
138.62
131.17
109.92
137.09
136.09
132.40
128.07
161.83
169.45
140.75
131.80
125.46
170.01
118.24
118.37
161.49
116.54
159.14
126.94
135.61
127.11
117.97
136.71
130.65
142.46
137.73
141.65
134.39
123.23
145.75
114.68
155.24
122.64
127.72
150.80
118.87
125.85
160.44
128.95
126.79
163.68
142.04
127.43
125.30
153.14
112.84
121.18
124.98
125.11
106.17
145.21
133.50
110.12
145.60
128.01
118.01
100.53
150.05
127.82
132.24
117.09
119.89
128.57
138.72
131.27
110.02
137.19
136.19
132.50
128.17
161.93
169.55
140.85
131.90
125.56
170.11
118.34
118.47
161.59
116.64
159.24
127.04
135.71
127.21
118.07
136.81
130.75
142.56
137.83
141.75
134.49
123.33
145.85
114.78
155.34
122.74
127.82
150.90
118.97
125.95
160.54
129.05
126.89
163.78
142.14
127.53
125.40
153.24
112.94
121.28
125.08
125.21
106.27
145.31
133.60
110.22
145.70
128.11
118.11
100.63
150.15
127.92
132.34
117.19
119.99
128.67
138.82
131.37
110.12
137.29
136.29
132.60
128.27
162.03
169.65
140.95
132.00
125.66
170.21
118.44
118.57
161.69
116.74
159.34
127.14
135.81
127.31
118.17
136.91
130.85
142.66
137.93
141.85
134.59
123.43
145.95
114.88
155.44
122.84
127.92
151.00
119.07
126.05
160.64
129.15
126.99
163.88
142.24
127.63
125.50
153.34
113.04
121.38
125.18
125.31
106.37
145.41
133.70
110.32
145.80
128.21
118.21
100.73
150.25
128.02
132.44
117.29
120.09
128.77
138.92
131.47
110.22
137.39
136.39
132.70
128.37
162.13
169.75
141.05
132.10
125.76
170.31
118.54
118.67
161.79
116.84
159.44
127.24
135.91
127.41
118.27
137.01
130.95
142.76
138.03
141.95
134.69
123.53
146.05
114.98
155.54
122.94
128.02
151.10
119.17
126.15
160.74
129.25
127.09
163.98
142.34
127.73
125.60
153.44
113.14
121.48
125.28
125.41
106.47
145.51
133.80
110.42
145.90
128.31
118.31
100.83
150.35
128.12
132.54
117.39
120.19
128.87
139.02
131.57
110.32
137.49
136.49
132.80
128.47
162.23
169.85
141.15
132.20
125.86
170.41
118.64
118.77
161.89
116.94
159.54
127.34
136.01
127.51
118.37
137.11
131.05
142.86
138.13
142.05
134.79
123.63
146.15
115.08
155.64
123.04
128.12
151.20
119.27
126.25
160.84
129.35
127.19
164.08
142.44
127.83
125.70
153.54
113.24
121.58
125.38
125.51
106.57
145.61
133.90
110.52
146.00
128.41
118.41
100.93
150.45
128.22
132.64
117.49
120.29
128.97
139.12
131.67
110.42
137.59
136.59
132.90
128.57
162.33
169.95
141.25
132.30
125.96
170.51
118.74
118.87
161.99
117.04
159.64
127.44
136.11
127.61
252y0611s1 10/3/06 (Open in ‘Print Layout’ format)
71
72
120.11
112.50
120.21
112.60
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
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55
56
57
58
59
s10
140.60
131.61
183.53
121.02
119.77
150.44
108.30
118.47
137.21
131.15
142.96
138.23
142.15
134.89
123.73
146.25
115.18
155.74
123.14
128.22
151.30
119.37
126.35
160.94
129.45
127.29
164.18
142.54
127.93
125.80
153.64
113.34
121.68
125.48
125.61
106.67
145.71
134.00
110.62
146.10
128.51
118.51
101.03
150.55
128.32
132.74
117.59
120.39
129.07
139.22
131.77
110.52
137.69
136.69
133.00
128.67
162.43
170.05
141.35
s0sq
19488.2
17059.0
33317.2
14404.8
14106.3
22332.3
11513.3
13799.2
18553.2
16939.0
20152.6
18832.1
19923.3
17926.5
15062.7
21097.6
13037.1
23944.5
14918.2
16184.9
22590.1
14011.5
15712.6
25580.8
16499.4
15949.2
26627.7
20033.6
16111.2
15575.0
23299.0
12620.3
14563.7
15495.3
15527.7
11166.1
20941.0
17689.0
12016.5
21054.0
16258.8
13808.6
10006.0
22365.2
16210.4
17355.4
13593.2
14254.0
16401.9
19104.8
17100.8
11994.6
18684.2
18411.8
17424.0
16299.6
26059.6
28577.9
19698.1
120.31
112.70
120.41
112.80
120.51
112.90
120.61
113.00
120.71
113.10
120.81
113.20
120.91
113.30
252y0611s1 10/3/06 (Open in ‘Print Layout’ format)
60
61
62
63
64
65
66
67
68
69
70
71
72
132.40
126.06
170.61
118.84
118.97
162.09
117.14
159.74
127.54
136.21
127.71
121.01
113.40
17266.0
15640.0
28767.6
13886.3
13916.9
25950.0
13488.5
25198.4
16012.4
18281.7
16055.4
14402.4
12633.8
MTB > describe c1
Descriptive Statistics: s0
Variable
s0
N
72
N*
0
Variable
s0
Maximum
182.53
Mean
132.34
SE Mean
1.97
MTB > describe c2 - c11
StDev
16.69
Minimum
100.03
Q1
120.01
Median
128.26
Q3
141.44
Shows means and standard deviations for Versions 1-10.
Descriptive Statistics: s1, s2, s3, s4, s5, s6, s7, s8, s9, s10
Variable
s1
s2
s3
s4
s5
s6
s7
s8
s9
s10
N
72
72
72
72
72
72
72
72
72
72
N*
0
0
0
0
0
0
0
0
0
0
Variable
s1
s2
s3
s4
s5
s6
s7
s8
s9
s10
Maximum
182.63
182.73
182.83
182.93
183.03
183.13
183.23
183.33
183.43
183.53
Mean
132.44
132.54
132.64
132.74
132.84
132.94
133.04
133.14
133.24
133.34
SE Mean
1.97
1.97
1.97
1.97
1.97
1.97
1.97
1.97
1.97
1.97
StDev
16.69
16.69
16.69
16.69
16.69
16.69
16.69
16.69
16.69
16.69
Minimum
100.13
100.23
100.33
100.43
100.53
100.63
100.73
100.83
100.93
101.03
Q1
120.11
120.21
120.31
120.41
120.51
120.61
120.71
120.81
120.91
121.01
Median
128.36
128.46
128.56
128.66
128.76
128.86
128.96
129.06
129.16
129.26
Q3
141.54
141.64
141.74
141.84
141.94
142.04
142.14
142.24
142.34
142.44
MTB > Save "C:\Documents and Settings\RBOVE\My Documents\Minitab\252x06101.MTW";
SUBC>
Replace.
Saving file as: 'C:\Documents and Settings\RBOVE\My
Documents\Minitab\252x061-01.MTW'
Existing file replaced.
252y0611s1 10/3/06 (Open in ‘Print Layout’ format)
H 0 :   128
The test done by this command is 
at the 99% level.
H 0 :   128
MTB > onet c1;
SUBC> test 128;
SUBC> conf 99.0;
SUBC> alter 1.
One-Sample T: s0
The results here are p-values between .004 and .015. Those that are below 1%
will lead to a rejection of the null hypothesis. 99% lower bound is the bottom of
a 1-sided confidence interval. Your t-ratio and standard error are also given.
Test of mu = 128 vs > 128
Variable
s0
MTB >
SUBC>
SUBC>
SUBC>
N
72
Mean
132.339
StDev
16.692
SE Mean
1.967
99%
Lower
Bound
127.657
T
2.21
P
0.015
SE Mean
1.967
99%
Lower
Bound
127.757
T
2.26
P
0.014
SE Mean
1.967
99%
Lower
Bound
127.857
T
2.31
P
0.012
SE Mean
1.967
99%
Lower
Bound
127.957
T
2.36
P
0.011
SE Mean
1.967
99%
Lower
Bound
128.057
T
2.41
P
0.009
onet c2;
test 128;
conf 99.0;
alter 1.
One-Sample T: s1
Test of mu = 128 vs > 128
Variable
s1
MTB >
SUBC>
SUBC>
SUBC>
N
72
Mean
132.439
StDev
16.692
onet c3;
test 128;
conf 99.0;
alter 1.
One-Sample T: s2
Test of mu = 128 vs > 128
Variable
s2
MTB >
SUBC>
SUBC>
SUBC>
N
72
Mean
132.539
StDev
16.692
onet c4;
test 128;
conf 99.0;
alter 1.
One-Sample T: s3
Test of mu = 128 vs > 128
Variable
s3
MTB >
SUBC>
SUBC>
SUBC>
N
72
Mean
132.639
StDev
16.692
onet c5;
test 128;
conf 99.0;
alter 1.
One-Sample T: s4
Test of mu = 128 vs > 128
Variable
s4
N
72
Mean
132.739
StDev
16.692
252y0611s1 10/3/06 (Open in ‘Print Layout’ format)
MTB >
SUBC>
SUBC>
SUBC>
onet c6;
test 128;
conf 99.0;
alter 1.
One-Sample T: s5
Test of mu = 128 vs > 128
Variable
s5
MTB >
SUBC>
SUBC>
SUBC>
N
72
Mean
132.839
StDev
16.692
SE Mean
1.967
99%
Lower
Bound
128.157
T
2.46
P
0.008
SE Mean
1.967
99%
Lower
Bound
128.257
T
2.51
P
0.007
SE Mean
1.967
99%
Lower
Bound
128.357
T
2.56
P
0.006
SE Mean
1.967
99%
Lower
Bound
128.457
T
2.61
P
0.005
SE Mean
1.967
99%
Lower
Bound
128.557
T
2.66
P
0.005
onet c7;
test 128;
conf 99.0;
alter 1.
One-Sample T: s6
Test of mu = 128 vs > 128
Variable
s6
MTB >
SUBC>
SUBC>
SUBC>
N
72
Mean
132.939
StDev
16.692
onet c8;
test 128;
conf 99.0;
alter 1.
One-Sample T: s7
Test of mu = 128 vs > 128
Variable
s7
MTB >
SUBC>
SUBC>
SUBC>
N
72
Mean
133.039
StDev
16.692
onet c9;
test 128;
conf 99.0;
alter 1.
One-Sample T: s8
Test of mu = 128 vs > 128
Variable
s8
MTB >
SUBC>
SUBC>
SUBC>
N
72
Mean
133.139
StDev
16.692
onet c10;
test 128;
conf 99.0;
alter 1.
One-Sample T: s9
Test of mu = 128 vs > 128
Variable
s9
N
72
Mean
133.239
StDev
16.692
252y0611s1 10/3/06 (Open in ‘Print Layout’ format)
MTB >
SUBC>
SUBC>
SUBC>
onet c11;
test 128;
conf 99.0;
alter 1.
One-Sample T: s10
Test of mu = 128 vs > 128
Variable
s10
N
72
Mean
133.339
StDev
16.692
MTB > Sort c1 c21;
SUBC>
MTB >
SUBC>
MTB >
SUBC>
MTB >
SUBC>
MTB >
SUBC>
MTB >
SUBC>
MTB >
SUBC>
MTB >
SUBC>
MTB >
SUBC>
MTB >
SUBC>
MTB >
SUBC>
MTB >
T
2.71
P
0.004
Now I am sorting the columns to place items in order by size.
This is used to find the confidence interval for the median and
to check the number of items below the alleged median.
By c1.
Sort c2 c22;
By c2.
Sort c3 c23;
By c3.
Sort c4 c24;
By c4.
Sort c5 c25;
By c5.
Sort c6 c26;
By c6.
Sort c7 c27;
By c7.
Sort c8 c28;
By c8.
Sort c9 c29;
By c9.
Sort c10 c30;
By c10.
Sort c11 c31;
By c11.
print c21 - c31
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
SE Mean
1.967
99%
Lower
Bound
128.657
so0
100.03
105.67
107.30
109.52
109.62
112.34
112.40
114.18
116.14
116.59
117.47
117.51
117.84
117.97
118.37
118.77
119.39
120.01
120.02
120.68
122.14
122.73
This is the data in order for Versions 0-1.
so1
100.13
105.77
107.40
109.62
109.72
112.44
112.50
114.28
116.24
116.69
117.57
117.61
117.94
118.07
118.47
118.87
119.49
120.11
120.12
120.78
122.24
122.83
so2
100.23
105.87
107.50
109.72
109.82
112.54
112.60
114.38
116.34
116.79
117.67
117.71
118.04
118.17
118.57
118.97
119.59
120.21
120.22
120.88
122.34
122.93
so3
100.33
105.97
107.60
109.82
109.92
112.64
112.70
114.48
116.44
116.89
117.77
117.81
118.14
118.27
118.67
119.07
119.69
120.31
120.32
120.98
122.44
123.03
so4
100.43
106.07
107.70
109.92
110.02
112.74
112.80
114.58
116.54
116.99
117.87
117.91
118.24
118.37
118.77
119.17
119.79
120.41
120.42
121.08
122.54
123.13
so5
100.53
106.17
107.80
110.02
110.12
112.84
112.90
114.68
116.64
117.09
117.97
118.01
118.34
118.47
118.87
119.27
119.89
120.51
120.52
121.18
122.64
123.23
so6
100.63
106.27
107.90
110.12
110.22
112.94
113.00
114.78
116.74
117.19
118.07
118.11
118.44
118.57
118.97
119.37
119.99
120.61
120.62
121.28
122.74
123.33
so7
100.73
106.37
108.00
110.22
110.32
113.04
113.10
114.88
116.84
117.29
118.17
118.21
118.54
118.67
119.07
119.47
120.09
120.71
120.72
121.38
122.84
123.43
so8
100.83
106.47
108.10
110.32
110.42
113.14
113.20
114.98
116.94
117.39
118.27
118.31
118.64
118.77
119.17
119.57
120.19
120.81
120.82
121.48
122.94
123.53
252y0611s1 10/3/06 (Open in ‘Print Layout’ format)
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
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
124.48
124.61
124.80
125.06
125.35
126.29
126.54
126.71
126.93
127.22
127.32
127.51
127.67
128.07
128.45
130.15
130.61
130.77
131.40
131.74
132.00
133.00
133.89
135.21
135.69
136.21
136.69
137.23
138.22
139.60
140.35
141.15
141.54
141.96
144.71
145.10
145.25
149.44
149.55
150.30
152.64
154.74
158.74
159.94
161.09
161.43
163.18
169.05
169.61
182.53
124.58
124.71
124.90
125.16
125.45
126.39
126.64
126.81
127.03
127.32
127.42
127.61
127.77
128.17
128.55
130.25
130.71
130.87
131.50
131.84
132.10
133.10
133.99
135.31
135.79
136.31
136.79
137.33
138.32
139.70
140.45
141.25
141.64
142.06
144.81
145.20
145.35
149.54
149.65
150.40
152.74
154.84
158.84
160.04
161.19
161.53
163.28
169.15
169.71
182.63
Row
1
2
3
4
5
6
7
8
9
10
11
so9
100.93
106.57
108.20
110.42
110.52
113.24
113.30
115.08
117.04
117.49
118.37
so10
101.03
106.67
108.30
110.52
110.62
113.34
113.40
115.18
117.14
117.59
118.47
124.68
124.81
125.00
125.26
125.55
126.49
126.74
126.91
127.13
127.42
127.52
127.71
127.87
128.27
128.65
130.35
130.81
130.97
131.60
131.94
132.20
133.20
134.09
135.41
135.89
136.41
136.89
137.43
138.42
139.80
140.55
141.35
141.74
142.16
144.91
145.30
145.45
149.64
149.75
150.50
152.84
154.94
158.94
160.14
161.29
161.63
163.38
169.25
169.81
182.73
124.78
124.91
125.10
125.36
125.65
126.59
126.84
127.01
127.23
127.52
127.62
127.81
127.97
128.37
128.75
130.45
130.91
131.07
131.70
132.04
132.30
133.30
134.19
135.51
135.99
136.51
136.99
137.53
138.52
139.90
140.65
141.45
141.84
142.26
145.01
145.40
145.55
149.74
149.85
150.60
152.94
155.04
159.04
160.24
161.39
161.73
163.48
169.35
169.91
182.83
124.88
125.01
125.20
125.46
125.75
126.69
126.94
127.11
127.33
127.62
127.72
127.91
128.07
128.47
128.85
130.55
131.01
131.17
131.80
132.14
132.40
133.40
134.29
135.61
136.09
136.61
137.09
137.63
138.62
140.00
140.75
141.55
141.94
142.36
145.11
145.50
145.65
149.84
149.95
150.70
153.04
155.14
159.14
160.34
161.49
161.83
163.58
169.45
170.01
182.93
124.98
125.11
125.30
125.56
125.85
126.79
127.04
127.21
127.43
127.72
127.82
128.01
128.17
128.57
128.95
130.65
131.11
131.27
131.90
132.24
132.50
133.50
134.39
135.71
136.19
136.71
137.19
137.73
138.72
140.10
140.85
141.65
142.04
142.46
145.21
145.60
145.75
149.94
150.05
150.80
153.14
155.24
159.24
160.44
161.59
161.93
163.68
169.55
170.11
183.03
125.08
125.21
125.40
125.66
125.95
126.89
127.14
127.31
127.53
127.82
127.92
128.11
128.27
128.67
129.05
130.75
131.21
131.37
132.00
132.34
132.60
133.60
134.49
135.81
136.29
136.81
137.29
137.83
138.82
140.20
140.95
141.75
142.14
142.56
145.31
145.70
145.85
150.04
150.15
150.90
153.24
155.34
159.34
160.54
161.69
162.03
163.78
169.65
170.21
183.13
125.18
125.31
125.50
125.76
126.05
126.99
127.24
127.41
127.63
127.92
128.02
128.21
128.37
128.77
129.15
130.85
131.31
131.47
132.10
132.44
132.70
133.70
134.59
135.91
136.39
136.91
137.39
137.93
138.92
140.30
141.05
141.85
142.24
142.66
145.41
145.80
145.95
150.14
150.25
151.00
153.34
155.44
159.44
160.64
161.79
162.13
163.88
169.75
170.31
183.23
125.28
125.41
125.60
125.86
126.15
127.09
127.34
127.51
127.73
128.02
128.12
128.31
128.47
128.87
129.25
130.95
131.41
131.57
132.20
132.54
132.80
133.80
134.69
136.01
136.49
137.01
137.49
138.03
139.02
140.40
141.15
141.95
142.34
142.76
145.51
145.90
146.05
150.24
150.35
151.10
153.44
155.54
159.54
160.74
161.89
162.23
163.98
169.85
170.41
183.33
252y0611s1 10/3/06 (Open in ‘Print Layout’ format)
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
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
118.41
118.74
118.87
119.27
119.67
120.29
120.91
120.92
121.58
123.04
123.63
125.38
125.51
125.70
125.96
126.25
127.19
127.44
127.61
127.83
128.12
128.22
128.41
128.57
128.97
129.35
131.05
131.51
131.67
132.30
132.64
132.90
133.90
134.79
136.11
136.59
137.11
137.59
138.13
139.12
140.50
141.25
142.05
142.44
142.86
145.61
146.00
146.15
150.34
150.45
151.20
153.54
155.64
159.64
160.84
161.99
162.33
164.08
169.95
170.51
183.43
MTB >
118.51
118.84
118.97
119.37
119.77
120.39
121.01
121.02
121.68
123.14
123.73
125.48
125.61
125.80
126.06
126.35
127.29
127.54
127.71
127.93
128.22
128.32
128.51
128.67
129.07
129.45
131.15
131.61
131.77
132.40
132.74
133.00
134.00
134.89
136.21
136.69
137.21
137.69
138.23
139.22
140.60
141.35
142.15
142.54
142.96
145.71
146.10
146.25
150.44
150.55
151.30
153.64
155.74
159.74
160.94
162.09
162.43
164.18
170.05
170.61
183.53
252y0611s1 10/3/06 (Open in ‘Print Layout’ format)
MTB > print c1
Data Display
s0
139.60
136.21
114.18
128.45
120.68
127.51
128.07
161.43
161.09
130.61
130.15
154.74
126.29
124.48
117.51
138.22
169.05
116.14
182.53
141.96
122.14
163.18
124.61
100.03
130.77
140.35
158.74
120.02
137.23
127.22
141.54
105.67
149.55
109.52
131.40
126.54
118.77
141.15
150.30
126.93
144.71
127.32
136.69
125.06
135.21
149.44
133.89
118.37
124.80
133.00
131.74
135.69
169.61
126.71
107.30
122.73
125.35
152.64
109.62
116.59
132.00
117.84
120.01
117.47
145.25
159.94
112.34
145.10
119.39
127.67
117.97
112.40
MTB > sum c1
Sum of s0
Sum of s0 = 9528.41
MTB > ssq c1
Sum of Squares of s0
Sum of squares (uncorrected) of s0 = 1280764
MTB > sum c13
Sum of C13
Sum of C13 = 12807637
 x  9528 .41 ,  x  1280763 .7 ,
 x  9528 .41  132 .339
x
2
n
sx2
x

n  72
72
2
 nx 2

1280763 .7  72132 .339 2
71
n 1
19783 .71

 278 .6439
71
The formula for the sample standard deviation is in Table 20 of the Supplement.
sx  278 .6439  16 .6926
 x  x 
2
MTB > mean c1
# Now I am using the definitional method. sx2 
n 1
Mean of s0
Mean of s0 = 132.339
MTB > let c14 = c1 - 132.339
C14 is now x0  x  . This is the same for all versions.
MTB > sum c14
Sum of s0adj
Sum of s0adj = 0.00200000
MTB > let c15 = c14 * c14
MTB > sum c15
This is
C15 is now x0  x 
2
Sum of s0adjsq
Sum of s0adjsq = 19783.2
 x  x  . This is the same for all versions.
This is
 x  x 
2
. This is the same for all versions.
.
252y0611s1 10/3/06 (Open in ‘Print Layout’ format)
MTB > let k1 = 19783.2/71
MTB > print k1
I am now dividing by n-1.
Data Display
K1
278.637
# This is the sample variance.
MTB > print c14 c15
Data Display
Row
s0adj
s0adjsq
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
40
41
42
43
44
45
46
47
48
49
50
7.261
-1.729
50.191
-12.319
-13.569
17.101
-25.039
-14.869
3.871
-2.189
9.621
4.891
8.811
1.551
-9.609
12.911
-18.159
22.401
-10.199
-5.119
17.961
-13.969
-6.989
27.601
-3.889
-6.049
30.841
9.201
-5.409
-7.539
20.301
-19.999
-11.659
-7.859
-7.729
-26.669
12.371
0.661
-22.719
12.761
-4.829
-14.829
-32.309
17.211
-5.019
-0.599
-15.749
-12.949
-4.269
5.881
52.72
2.99
2519.14
151.76
184.12
292.44
626.95
221.09
14.98
4.79
92.56
23.92
77.63
2.41
92.33
166.69
329.75
501.80
104.02
26.20
322.60
195.13
48.85
761.82
15.12
36.59
951.17
84.66
29.26
56.84
412.13
399.96
135.93
61.76
59.74
711.24
153.04
0.44
516.15
162.84
23.32
219.90
1043.87
296.22
25.19
0.36
248.03
167.68
18.22
34.59
#These are correct for all versions. First column shows x  x .
Second column shows x  x 2 .
252y0611s1 10/3/06 (Open in ‘Print Layout’ format)
51
52
53
54
55
56
57
58
59
60
61
62
63
64
65
66
67
68
69
70
71
72
-1.569
-22.819
4.351
3.351
-0.339
-4.669
29.091
36.711
8.011
-0.939
-7.279
37.271
-14.499
-14.369
28.751
-16.199
26.401
-5.799
2.871
-5.629
-12.329
-19.939
2.46
520.71
18.93
11.23
0.11
21.80
846.29
1347.70
64.18
0.88
52.98
1389.13
210.22
206.47
826.62
262.41
697.01
33.63
8.24
31.69
152.00
397.56