Design of Experiments workshop # 1Simple 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 ValuesNonsmokers Smokers

0.97 1.16 0.480.72 0.86 0.71

1 0.85 0.980.81 0.58 0.680.62 0.57 1.181.32 0.64 1.361.24 0.98 0.780.99 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.

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10.41 A study was conducted by the Department 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

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Mogul Corporation settling pond enter the stream near 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 MeterStation 1 Station 2

5030 4980 2800 281013,700 11,910 4670 133010,730 8130 6890 332011,400 26,850 7720 1230

860 17,660 7030 21302200 22,800 7330 21904250 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

N Mean StDev SE Meanestación1 16 9898 7874 1969estación2 12 4121 2480 716

Difference = mu (estación1) - mu (estación2)Estimate for difference: 5776.6795% CI for difference: (1375.93, 10177.41)T-Test of difference = 0 (vs not =): T-Value = 2.76 P-Value = 0.013 DF = 18

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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 LiterCar Radial Tires Belted Tires

1 4.2 4.12 4.7 4.93 6.6 6.24 7 6.95 6.7 6.86 4.5 4.47 5.7 5.78 6 5.89 7.4 6.9

10 4.9 4.711 6.1 612 5.2 4.9

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Can we conclude that cars equipped with radial tires give better fuel economy than those equipped with belted tires? Assume the populations to be normally distributed. Use a P-value in your conclusion.

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Two-sample T for Radial Tires vs Belted Tires

N Mean StDev SE MeanRadial Tires 12 5.75 1.05 0.30Belted Tires 12 5.608 0.994 0.29

Difference = mu (Radial Tires) - mu (Belted Tires)

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Estimate for difference: 0.14166795% 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 DifferencesSubject Prefatigue Postfatigue

1 158 912 92 593 65 2154 98 2265 33 2236 89 917 148 928 58 1779 142 134

10 117 11611 74 15312 66 21913 109 14314 57 16415 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.

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Paired T for Prefatigue - Postfatigue

N Mean StDev SE MeanPrefatigue 15 92.733 36.433 9.407Postfatigue 15 146.867 55.707 14.383Difference 15 -54.1333 83.0025 21.4311

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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

N Mean StDev SE MeanPrefatigue 15 92.7 36.4 9.4Postfatigue 15 146.9 55.7 14

Difference = mu (Prefatigue) - mu (Postfatigue)Estimate for difference: -54.133395% CI for difference: (-89.6045, -18.6622)

T-Test of difference = 0 (vs not =): T-Value = -3.15 P-Value = 0.004 DF = 24

10.44In 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 HamSlice Before Storage After Storage

1 224 1162 270 963 400 2394 444 3295 590 4376 660 5977 1400 6898 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?

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Two-sample T for Before Storage vs After Storage

N Mean StDev SE MeanBefore Storage 8 584 371 131After Storage 8 385 226 80

Difference = mu (Before Storage) - mu (After Storage)Estimate for difference: 198.625

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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 Strength1 Hot 51201 Cold 82002 Hot 100002 Cold 86003 Hot 100003 Cold 92004 Hot 100004 Cold 62005 Hot 100005 Cold 100006 Hot 79006 Cold 52007 Hot 5107 Cold 8858 Hot 10208 Cold 460

(a) Write an appropriate hypothesis to determine if there is a significant difference in strength between the hot and cold incisions. (b) Test the hypothesis using a paired t-test. Use a P-value in your conclusion.

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Paired T for Dog - Strength

N Mean StDev SE MeanDog 16 4.50 2.37 0.59Strength 16 6455.94 3789.23 947.31Difference 16 -6451.44 3790.84 947.71

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

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