Design of Experiments workshop # 1 Simple comparison experiments
Name _______ ALEX MUÑOZ ARPAIZ ____________ ____________ Date __25/02/13____ Score _________ _________ 10.40
In a study conducted at Virginia Tech, the plasma ascorbic acid levels of pregnant women were compared for smokers versus nonsmokers. Thirty-two women in the last three months of pregnancy, free of major health disorders and ranging in age from 15 to 32 years, were selected for the study. Prior to the collection of 20 ml of blood, the participants were told to avoid breakfast, forgo their vitamin supplements, and avoid foods high in ascorbic acid content. From the blood samples, the following plasma ascorbic acid values were determined, in milligrams per 100 milliliters: Plasma Ascorbic Acid Values Nonsmokers
Smokers
0.97
1.16
0.48
0.72
0.86
0.71
1
0.85
0.98
0.81
0.58
0.68
0.62
0.57
1.18
1.32
0.64
1.36
1.24
0.98
0.78
0.99
1.09
1.64
0.9
0.92
0.74
0.78
0.88
1.24
0.94
1.18
Is there sufficient evidence to conclude that there is a difference between plasma ascorbic acid levels of smokers and nonsmokers? Assume that the two sets of data came from normal populations with unequal variances. Use a P-value.
1 Workshop 1. DOE
Probability Plot of Nonsmokers Normal 99
95 90
Mean StDev N AD P-Value
0.9158 0.2144 24 0.212 0.837
Mean StDev N AD P-Value
0.9763 0.3915 8 0.239 0.678
80 70
t n 60 e c 50 r e 40 P
30 20 10 5
1
0.50
0.75
1.00
1.25
1.50
Nonsmokers
Probability Plot of Smokers Normal 99
95 90 80 70
t n 60 e c 50 r e 40 P
30 20 10 5
1
0.0
0.5
1.0
1.5
2.0
Smokers
10.41
A study was conducted by the Departme nt of Zoology at Virginia Tech to determine if there is a significant difference in the density of organisms at two different stations located on Cedar Run, a secondary stream in the Roanoke River drainage basin. Sewage from a sewage treatment plant and overflow from the Federal
2 Workshop 1. DOE
Mogul Corporation settling pond enter the stream ne ar its headwaters. The following data give the density measurements, in number of organisms per square meter, at the two collecting stations:
Number of Organisms per Square Meter Station 1
Station 2
5030
4980
2800
2810
13,700
11,910
4670
1330
10,730
8130
6890
3320
11,400
26,850
7720
1230
860
17,660
7030
2130
2200
22,800
7330
2190
4250
1130
15,040
1690
Can we conclude, at the 0.05 level of significance, that the average densities at the two stations are equal? Assume that the observations come from normal populations with different variances. Two-sample T for estación1 vs estación2
estación1 estación2
N 16 12
Mean 9898 4121
StDev 7874 2480
SE Mean 1969 716
Difference = mu (estación1) - mu (estación2) Estimate for difference: 5776.67 95% CI for difference: (1375.93, 10177.41) T-Test of difference = 0 (vs not =): T-Value = 2.76
P-Value = 0.013
DF = 18
Probability Plot of estación1 Normal 99
Mean S tDev N A D P-Value
95 90
9898 7874 16 0.413 0.297
80 70
t n 60 e c 50 r e 40 P
30 20 10 5
1
-10000
0
10000
20000
30000
estación1
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Probability Plot of estación2 Normal 99
Mean S tD ev N AD P-Value
95 90
4121 2480 12 0.698 0.050
80 70
t n 60 e c 50 r e 40 P
30 20 10 5
1
0
2500
5000
7500
10000
estación2
10.45
A taxi company manager is trying to decide whether the use of radial tires instead of regular belted tires improves fuel economy. Twelve cars were equipped with radial tires and driven over a prescribed test course. Without changing drivers, the same cars were then equipped with regular belted tires and driven once again over the test course. The gasoline consumption, in kilometers per liter, was recorded as follows:
Kilometers per Liter Car
Radial Tires Belted Tires 1
4.2
4.1
2
4.7
4.9
3
6.6
6.2
4
7
6.9
5
6.7
6.8
6
4.5
4.4
7
5.7
5.7
8
6
5.8
9
7.4
6.9
10
4.9
4.7
11
6.1
6
12
5.2
4.9
4 Workshop 1. DOE
Can we conclude that cars equipped with radial tires give better fuel e conomy than those equipped with belted tires? Assume the populations to be normally distributed. Use a P-value in your conclusion.
Probability Plot of Radial Tires Normal 99
Mean StDev N A D P-Value
95 90
5.75 1.053 12 0.222 0.780
80 70
t n 60 e c 50 r e 40 P
30 20 10 5
1
3
4
5
6
7
8
Radial Tires
Probability Plot of Belted Tires Normal 99 Mean StDev N AD P-Value
95 90
5.608 0.9940 12 0.348 0.414
80 70
t n 60 e c 50 r e 40 P
30 20 10 5
1
3
4
5
6
7
8
Belted Tires
5 Workshop 1. DOE
Summary for Radial Tires A nderson-Darling N ormality Test
4
5
6
A -Squared P -V alue
0.22 0.780
M ean S tD ev V ariance S k ew n es s Kurtosis N
5.7500 1.0527 1.1082 0 .03 366 -1.30654 12
M inim um 1st Q uartile M edian 3rd Quartile M a xim um
7
4.2000 4.7500 5.8500 6.6750 7. 4000
95% C onfidence Interv al for Mean 5.0811
6.4189
95% C onfidence Interv al for Median 4.7526
6.6737
95% C onfidence Interv al for StD ev
95% Confidence Intervals
0.7457
1.7874
Mean Median 4.5
5.0
5.5
6.0
6.5
Summary for Belted Tires A nderson-Darling N ormality Test
4.0
4.5
5.0
5.5
6.0
6.5
7.0
A -Squared P -V alue
0.35 0.414
M ean S tD ev V ariance Skewness Kurtosis N
5.6083 0.9940 0.9881 -0.04206 -1.41504 12
M inim um 1st Q uartile M edian 3rd Quartile M a xim um
4.1000 4.7500 5.7500 6.6500 6. 9000
95% C onfidence Interv al for Mean 4.9768
6.2399
95% C onfidence Interv al for Median 4.7526
6.6421
95% C onfidence Interv al for StD ev
95% Confidence Intervals
0.7042
1.6878
Mean Median 4.5
5.0
5.5
6.0
6.5
Two-sample T for Radial Tires vs Belted Tires
Radial Tires Belted Tires
N 12 12
Mean 5.75 5.608
StDev 1.05 0.994
SE Mean 0.30 0.29
Difference = mu (Radial Tires) - mu (Belted Tires)
6 Workshop 1. DOE
Estimate for difference: 0.141667 95% CI for difference: (-0.727529, 1.010862) T-Test of difference = 0 (vs not =): T-Value = 0.34
P-Value = 0.738
DF = 21
10.43
According to published reports, practice under fatigued conditions distorts mechanisms that govern performance. An experiment was conducted using 15 college males, who were trained to make a continuous horizontal right-to-left arm movement from a micro switch to a barrier, knocking over the barrier coincident with the arrival of a clock sweep hand to the 6 o’clock position. The absolute value of the difference between the time, in milliseconds, that it took to knock over the barrier and the time for the sweep hand to reach the 6 o’clock position (500 msec) was recorded. Each participant performed the task five times under prefatigue and postfatigue conditions, and the sums of the absolute differences for the five performances were recorded.
Absolute Time Differences Subject Prefatigue Postfatigue 1
158
91
2
92
59
3
65
215
4
98
226
5
33
223
6
89
91
7
148
92
8
58
177
9
142
134
10
117
116
11
74
153
12
66
219
13
109
143
14
57
164
15
85
100
An increase in the mean absolute time difference when the task is performed under postfatigue conditions would support the claim that practice under fatigued conditions distorts mechanisms that govern performance. Assuming the populations to be normally distributed, test this claim.
7 Workshop 1. DOE
Probability Plot of Prefatigue Normal 99
95 90
Mean StDev N A D P-Value
92.73 36.43 15 0.279 0.594
Mean StDev N A D P-Value
146.9 55.71 15 0.426 0.274
80 70
t n 60 e c 50 r e 40 P
30 20 10 5
1
0
50
100
150
200
Prefatigue
Probability Plot of Postfatigue Normal 99
95 90 80 70
t n 60 e c 50 r e 40 P
30 20 10 5
1
0
50
100
150
200
250
300
Postfatigue
Paired T for Prefatigue - Postfatigue
Prefatigue Postfatigue Difference
N 15 15 15
Mean 92.733 146.867 -54.1333
StDev 36.433 55.707 83.0025
SE Mean 9.407 14.383 21.4311
8 Workshop 1. DOE
95% CI for mean difference: (-100.0986, -8.1681) T-Test of mean difference = 0 (vs not = 0): T-Value = -2.53
P-Value = 0.024
Two-Sample T-Test and CI: Prefatigue, Postfatigue Two-sample T for Prefatigue vs Postfatigue
Prefatigue Postfatigue
N 15 15
Mean 92.7 146.9
StDev 36.4 55.7
SE Mean 9.4 14
Difference = mu (Prefatigue) - mu (Postfatigue) Estimate for difference: -54.1333 95% CI for difference: (-89.6045, -18.6622)
T-Test of difference = 0 (vs not =): T-Value = -3.15 P-Value = 0.004 DF = 24
10.44
In a study conducted by the Department of Human Nutrition and Foods at Virginia Tech, the following data were recorded on sorbic acid residuals, in parts per million, in ham immediately after dipping in a sorbate solution and after 60 days of storage: Sorbic Acid Residual in Ham Slice
Before Storage After Storage 1
224
116
2
270
96
3
400
239
4
444
329
5
590
437
6
660
597
7
1400
689
8
680
576
Assuming the populations to be normally distributed, is there sufficient evidence, at the 0.05 level of significance, to say that the length of storage influences sorbic acid residual concentrations?
9 Workshop 1. DOE
Probability Plot of Before Storage Normal 99
95 90
Mean StDev N A D P-Value
583.5 370.8 8 0.577 0.089
Mean StDev N A D P-Value
384.9 225.8 8 0.257 0.612
80 70
t n 60 e c 50 r e 40 P
30 20 10 5
1
-500
0
500
1000
1500
Before Storage
Probability Plot of After Storage Normal 99
95 90 80 70
t n 60 e c 50 r e 40 P
30 20 10 5
1
0
250
500
750
1000
After Storage
10 Workshop 1. DOE
Summary for Before Storage A nderson-Darling N ormality Test
200
400
600
800
1000
1200
A -Squared P -V alue
0.58 0.089
M ean S tD ev V ariance S k ew n es s K urtosis N
583.50 370.82 137504.86 1. 71659 3. 66704 8
M inimum 1 st Q u a rt il e M edian 3rd Quartile M a xim um
1400
224.00 3 02 .50 517.00 675.00 1400. 00
95% C onfidence Interv al for Mean 273.49
893.51
95% C onfidence Interv al for Median 267.04
726.34
95% C onfidence Interv al for StD ev
95% Confidence Intervals
245.17
754.71
Mean Median 300
400
500
600
700
800
900
Summary for After Storage A nderson-Darling N ormality Test
100
200
300
400
500
600
A -Squared P -V alue
0.26 0.612
M ean S tDev V ariance Skewness K urt os is N
384.88 225.79 50982.70 -0.03352 -1. 658 53 8
M inimum 1st Quartile M edian 3rd Q uartile M axim um
700
96.00 146.75 383.00 591.75 689.00
95% C onfidence Interv al for Mean 196.11
573.64
95% C onfidence Interv al for Median 114.71
602.92
95% C onfidence Interv al for StD ev
95% Confidence Intervals
149.29
459.55
Mean Median 100
200
300
400
500
600
Two-sample T for Before Storage vs After Storage
Before Storage After Storage
N 8 8
Mean 584 385
StDev 371 226
SE Mean 131 80
Difference = mu (Before Storage) - mu (After Storage) Estimate for difference: 198.625
11 Workshop 1. DOE
95% CI for difference: (-139.217, 536.467) T-Test of difference = 0 (vs not =): T-Value = 1.29
P-Value = 0.222
DF = 11
10.53
A study was conducted at the Department of Veterinary Medicine at Virginia Tech to determine if the “strength” of a wound from surgical incision is affected by the temperature of the knife. Eight dogs were used in the experiment. “Hot” and “cold” incisions were made on the abdomen of each dog, and the strength was measured. The resulting data appear below. Dog
Knife
Strength
1 Hot
5120
1 Cold
8200
2 Hot
10000
2 Cold
8600
3 Hot
10000
3 Cold
9200
4 Hot
10000
4 Cold
6200
5 Hot
10000
5 Cold
10000
6 Hot
7900
6 Cold
5200
7 Hot
510
7 Cold
885
8 Hot
1020
8 Cold
460
(a) Write an appropriate hypothesis to determine if there is a significant difference in strength betwe en the hot and cold incisions. (b) Test the hypothesis using a paired t -test. Use a P-value in your conclusion.
12 Workshop 1. DOE
Probability Plot of Strength Normal 99
Mean StDev N AD P-Value
95 90
6456 3789 16 1.117 <0.005
80 70
t n 60 e c 50 r e 40 P
30 20 10 5
1
-5000
0
5000
10000
15000
Strength
Summary for Strength A nderson-Darling N ormality Test
0
2000
4000
6000
8000
10000
A -Squa red P -V alue <
1.12 0.005
M ean S tDev V ariance S k ew n ess K urtosis N
6455.9 3789.2 14358244.1 -0. 70338 -1. 16444 16
M inimum 1 st Q u a rti le M edian 3rd Quartile M axim um
460.0 2 04 5. 0 8050.0 10000.0 10000.0
95% C onfidence Interv al for Mean 4436.8
8475.1
95% C onfidence Interval for Median 4147.2
10000.0
95% C onfidence Interval for StDev 95 % Confidence Intervals
2799.1
5864.6
Mean Median 4000
5000
6000
7000
8000
9000
10000
Paired T for Dog - Strength
Dog Strength Difference
N 16 16 16
Mean 4.50 6455.94 -6451.44
StDev 2.37 3789.23 3790.84
SE Mean 0.59 947.31 947.71
95% CI for mean difference: (-8471.43, -4431.44) T-Test of mean difference = 0 (vs not = 0): T-Value = -6.81
P-Value = 0.000
13 Workshop 1. DOE
