Exercise 9 - West Chester University's - On-line Web …courses.wcupa.edu/rbove/eco252/252y0821a.doc · Web viewSum of C7_Fat = 91 Sum of C7_Fit Sum of C7_Fit = 260 Mann-Whitney Test

252y0821a 3/31/08

Output for the last Take-home problemAbout 60 Pages – You may want to print only your problem.

————— 4/4/2008 9:04:29 PM ———————————————————— Welcome to Minitab, press F1 for help.MTB > WOpen "C:\Documents and Settings\RBOVE\My Documents\Minitab\252x0821-02V1.MTW". Version 1 – Only data printouts are shown.Retrieving worksheet from file: 'C:\Documents and Settings\RBOVE\MyDocuments\Minitab\252x0821-02V1.MTW'Worksheet was saved on Wed Apr 02 2008

Results for: 252x0821-02V1.MTWMTB > Execute "C:\Documents and Settings\RBOVE\My Documents\Minitab\252x0822.mtb" 1.Executing from file: C:\Documents and Settings\RBOVE\My Documents\Minitab\252x0822.mtb Descriptive Statistics: Fat, Fit, diff Variable N N* Mean SE Mean StDev Minimum Q1 Median Q3Fat 13 0 4.613 0.197 0.711 3.120 4.140 4.740 5.200Fit 13 0 6.411 0.123 0.442 5.710 6.100 6.380 6.605diff 13 0 -1.798 0.211 0.761 -2.960 -2.505 -2.000 -1.195

Variable MaximumFat 5.530Fit 7.320diff -0.400 Covariances: Fat, Fit Diagonal Items are Variances. Fat FitFat 0.505090Fit 0.060514 0.195241 Data Display Row Fat fatsq Fit fitsq diff diffsq 1 4.24 17.9776 6.42 41.2164 -2.18 4.7524 2 4.74 22.4676 7.32 53.5824 -2.58 6.6564 3 4.93 24.3049 6.38 40.7044 -1.45 2.1025 4 4.16 17.3056 6.16 37.9456 -2.00 4.0000 5 5.53 30.5809 5.93 35.1649 -0.40 0.1600 6 4.12 16.9744 7.08 50.1264 -2.96 8.7616 7 5.10 26.0100 6.37 40.5769 -1.27 1.6129 8 4.47 19.9809 6.53 42.6409 -2.06 4.2436 9 5.30 28.0900 6.68 44.6224 -1.38 1.9044 10 3.12 9.7344 5.71 32.6041 -2.59 6.7081 11 3.77 14.2129 6.20 38.4400 -2.43 5.9049 12 5.09 25.9081 6.04 36.4816 -0.95 0.9025 13 5.40 29.1600 6.52 42.5104 -1.12 1.2544 Sum of Fat Sum of Fat = 59.97 Sum of Squares of Fat Sum of squares (uncorrected) of Fat = 282.707

Sum of fatsq Sum of fatsq = 282.707

Sum of Fit Sum of Fit = 83.34

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Sum of Squares of Fit Version 1Sum of squares (uncorrected) of Fit = 536.616

Sum of fitsq Sum of fitsq = 536.616

Sum of diff Sum of diff = -23.37 Sum of Squares of diff Sum of squares (uncorrected) of diff = 48.9637 Sum of diffsq Sum of diffsq = 48.9637

Two-Sample T-Test and CI: Fat, Fit Two-sample T for Fat vs Fit N Mean StDev SE MeanFat 13 4.613 0.711 0.20Fit 13 6.411 0.442 0.12Difference = mu (Fat) - mu (Fit)Estimate for difference: -1.79895% upper bound for difference: -1.401T-Test of difference = 0 (vs <): T-Value = -7.75 P-Value = 0.000 DF = 24Both use Pooled StDev = 0.5917 Two-Sample T-Test and CI: Fat, Fit Two-sample T for Fat vs Fit N Mean StDev SE MeanFat 13 4.613 0.711 0.20Fit 13 6.411 0.442 0.12Difference = mu (Fat) - mu (Fit)Estimate for difference: -1.79895% upper bound for difference: -1.397T-Test of difference = 0 (vs <): T-Value = -7.75 P-Value = 0.000 DF = 20 Test for Equal Variances: Fat, Fit 95% Bonferroni confidence intervals for standard deviations N Lower StDev UpperFat 13 0.487243 0.710697 1.27091Fit 13 0.302933 0.441861 0.79016

F-Test (Normal Distribution)Test statistic = 2.59, p-value = 0.113Levene's Test (Any Continuous Distribution)Test statistic = 3.40, p-value = 0.078Test for Equal Variances for Fat, Fit All graphs are at end of printout.. Covariances: diff, diff Diagonal Items are Variances. diff diffdiff 0.579303diff 0.579303 0.579303 Probability Plot of diff Probability Plot of diff Probability Plot of diff

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Data Display Columns and Ranks for Wilcoxon-Mann-Whitney TestRow Fat C7_Fat Fit C7_Fit 1 4.24 5 6.42 21 2 4.74 7 7.32 26 3 4.93 8 6.38 20 4 4.16 4 6.16 17 5 5.53 13 5.93 15 6 4.12 3 7.08 25 7 5.10 10 6.37 19 8 4.47 6 6.53 23 9 5.30 11 6.68 24 10 3.12 1 5.71 14 11 3.77 2 6.20 18 12 5.09 9 6.04 16 13 5.40 12 6.52 22 Sum of C7_Fat Version 1Sum of C7_Fat = 91 Sum of C7_Fit Sum of C7_Fit = 260

Mann-Whitney Test and CI: Fat, Fit N MedianFat 13 4.7400Fit 13 6.3800Point estimate for ETA1-ETA2 is -1.750095.4 Percent CI for ETA1-ETA2 is (-2.2598,-1.2297)W = 91.0Test of ETA1 = ETA2 vs ETA1 < ETA2 is significant at 0.0000 Paired T-Test and CI: Fat, Fit Paired T for Fat - Fit N Mean StDev SE MeanFat 13 4.613 0.711 0.197Fit 13 6.411 0.442 0.123Difference 13 -1.798 0.761 0.21195% upper bound for mean difference: -1.421T-Test of mean difference = 0 (vs < 0): T-Value = -8.52 P-Value = 0.000 Wilcoxon Signed Rank Test: diff Test of median = 0.000000 versus median < 0.000000 N for Wilcoxon Estimated N Test Statistic P Mediandiff 13 13 0.0 0.001 -1.780

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Data Display Table for Wilcoxon Signed Rank Test. Version 1 Sgnd Last two columns give Signed Rank Row Fat Fit diff C11 rank C13 1 4.24 6.42 -2.18 2.18 9 -1 2 4.74 7.32 -2.58 2.58 11 -1 3 4.93 6.38 -1.45 1.45 6 -1 4 4.16 6.16 -2.00 2.00 7 -1 5 5.53 5.93 -0.40 0.40 1 -1 6 4.12 7.08 -2.96 2.96 13 -1 7 5.10 6.37 -1.27 1.27 4 -1 8 4.47 6.53 -2.06 2.06 8 -1 9 5.30 6.68 -1.38 1.38 5 -1 10 3.12 5.71 -2.59 2.59 12 -1 11 3.77 6.20 -2.43 2.43 10 -1 12 5.09 6.04 -0.95 0.95 2 -1 13 5.40 6.52 -1.12 1.12 3 -1

Data Display Table for Wilcoxon Signed Rank Test for Median of 2. Sgnd Last two columns give Signed RankRow diff C14 C15 C11 rank C13 1 -2.18 -2 -0.18 0.18 3 -1 2 -2.58 -2 -0.58 0.58 6 -1 3 -1.45 -2 0.55 0.55 5 1 4 -2.00 -2 0.00 0.00 1 0 5 -0.40 -2 1.60 1.60 13 1 6 -2.96 -2 -0.96 0.96 11 -1 7 -1.27 -2 0.73 0.73 9 1 8 -2.06 -2 -0.06 0.06 2 -1 9 -1.38 -2 0.62 0.62 8 1 10 -2.59 -2 -0.59 0.59 7 -1 11 -2.43 -2 -0.43 0.43 4 -1 12 -0.95 -2 1.05 1.05 12 1 13 -1.12 -2 0.88 0.88 10 1 Wilcoxon Signed Rank Test: C15 Test of median = 0.000000 versus median not = 0.000000 N for Wilcoxon Estimated N Test Statistic P MedianC15 13 12 51.0 0.367 0.2200 Wilcoxon Signed Rank Test: diff Test of median = -2.000 versus median not = -2.000 N for Wilcoxon Estimated N Test Statistic P Mediandiff 13 12 51.0 0.367 -1.780

Mean of C20 Mean of C20 = -1.79769 Standard Deviation of C21 Standard deviation of C21 = 0.761119

Descriptive Statistics: C22 Variable N N* Mean SE Mean StDev Minimum Q1 Median Q3C22 13 0 -0.000 0.277 1.000 -1.527 -0.929 -0.266 0.792

Variable MaximumC22 1.836

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Data Display Table for Lilliefors Test – Last Column is absolute D. Version 1Row dmoved dordered z_d Fe O cumO Fo Fe-Fo 1 -2.18 -2.96 -1.52710 0.063368 1 1 0.07692 0.013555 2 -2.58 -2.59 -1.04098 0.148943 1 2 0.15385 0.004903 3 -1.45 -2.58 -1.02784 0.152013 1 3 0.23077 0.078756 4 -2.00 -2.43 -0.83076 0.203055 1 4 0.30769 0.104638 5 -0.40 -2.18 -0.50230 0.307729 1 5 0.38462 0.076886 6 -2.96 -2.06 -0.34463 0.365185 1 6 0.46154 0.096354 7 -1.27 -2.00 -0.26580 0.395196 1 7 0.53846 0.143266 8 -2.06 -1.45 0.45682 0.676099 1 8 0.61538 0.060714 9 -1.38 -1.38 0.54879 0.708424 1 9 0.69231 0.016116 10 -2.59 -1.27 0.69331 0.755943 1 10 0.76923 0.013288 11 -2.43 -1.12 0.89039 0.813372 1 11 0.84615 0.032782 12 -0.95 -0.95 1.11374 0.867306 1 12 0.92308 0.055771 13 -1.12 -0.40 1.83636 0.966848 1 13 1.00000 0.033152 Maximum of Fe-Fo In table above Fe column is Exact Cumulative Probability. Maximum of Fe-Fo = 0.143266Note that p-values for all three tests of Normality are above 5% - We cannot reject Normality.

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MTB > WOpen "C:\Documents and Settings\RBOVE\My Documents\Minitab\252x0821-02V2.MTW". Version 2 – Only data printouts are shown.Retrieving worksheet from file: 'C:\Documents and Settings\RBOVE\MyDocuments\Minitab\252x0821-02V2.MTW'Worksheet was saved on Wed Apr 02 2008Results for: 252x0821-02V2.MTWMTB > exec "252x0822.mtb"Executing from file: 252x0822.mtb Descriptive Statistics: Fat, Fit, diff Variable N N* Mean SE Mean StDev Minimum Q1 Median Q3Fat 13 0 4.671 0.196 0.708 3.120 4.140 4.930 5.200Fit 13 0 6.431 0.124 0.448 5.710 6.100 6.380 6.680diff 13 0 -1.760 0.209 0.753 -2.960 -2.505 -1.690 -1.195

Variable MaximumFat 5.530Fit 7.320diff -0.400 Covariances: Fat, Fit Diagonal Items are Variances. Fat FitFat 0.501724Fit 0.068008 0.200841

Data Display Row Fat fatsq Fit fitsq diff diffsq 1 4.99 24.9001 6.68 44.6224 -1.69 2.8561 2 4.74 22.4676 7.32 53.5824 -2.58 6.6564 3 4.93 24.3049 6.38 40.7044 -1.45 2.1025 4 4.16 17.3056 6.16 37.9456 -2.00 4.0000 5 5.53 30.5809 5.93 35.1649 -0.40 0.1600 6 4.12 16.9744 7.08 50.1264 -2.96 8.7616 7 5.10 26.0100 6.37 40.5769 -1.27 1.6129 8 4.47 19.9809 6.53 42.6409 -2.06 4.2436 9 5.30 28.0900 6.68 44.6224 -1.38 1.9044 10 3.12 9.7344 5.71 32.6041 -2.59 6.7081 11 3.77 14.2129 6.20 38.4400 -2.43 5.9049 12 5.09 25.9081 6.04 36.4816 -0.95 0.9025 13 5.40 29.1600 6.52 42.5104 -1.12 1.2544 Sum of Fat Sum of Fat = 60.72 Sum of Squares of Fat Sum of squares (uncorrected) of Fat = 289.630 Sum of fatsq Sum of fatsq = 289.630 Sum of Fit Sum of Fit = 83.6

Sum of Squares of Fit Sum of squares (uncorrected) of Fit = 540.022


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Sum of diff Version 2Sum of diff = -22.88 Sum of Squares of diff Sum of squares (uncorrected) of diff = 47.0674 Sum of diffsq Sum of diffsq = 47.0674 Two-Sample T-Test and CI: Fat, Fit Two-sample T for Fat vs Fit N Mean StDev SE MeanFat 13 4.671 0.708 0.20Fit 13 6.431 0.448 0.12Difference = mu (Fat) - mu (Fit)Estimate for difference: -1.76095% upper bound for difference: -1.362T-Test of difference = 0 (vs <): T-Value = -7.57 P-Value = 0.000 DF = 24Both use Pooled StDev = 0.5927 Two-Sample T-Test and CI: Fat, Fit Two-sample T for Fat vs Fit N Mean StDev SE MeanFat 13 4.671 0.708 0.20Fit 13 6.431 0.448 0.12Difference = mu (Fat) - mu (Fit)Estimate for difference: -1.76095% upper bound for difference: -1.359T-Test of difference = 0 (vs <): T-Value = -7.57 P-Value = 0.000 DF = 20 Test for Equal Variances: Fat, Fit 95% Bonferroni confidence intervals for standard deviations N Lower StDev UpperFat 13 0.485617 0.708325 1.26667Fit 13 0.307247 0.448153 0.80142F-Test (Normal Distribution)Test statistic = 2.50, p-value = 0.127Levene's Test (Any Continuous Distribution)Test statistic = 1.58, p-value = 0.221Test for Equal Variances for Fat, Fit All graphs are at end of printout.

Covariances: diff, diff Diagonal Items are Variances. diff diffdiff 0.566550diff 0.566550 0.566550 Probability Plot of diff Probability Plot of diff Probability Plot of diff

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Data Display Columns and Ranks for Wilcoxon-Mann-Whitney TestRow Fat C7_Fat Fit C7_Fit 1 4.99 8 6.68 23.5 2 4.74 6 7.32 26.0 3 4.93 7 6.38 20.0 4 4.16 4 6.16 17.0 5 5.53 13 5.93 15.0 6 4.12 3 7.08 25.0 7 5.10 10 6.37 19.0 8 4.47 5 6.53 22.0 9 5.30 11 6.68 23.5 10 3.12 1 5.71 14.0 11 3.77 2 6.20 18.0 12 5.09 9 6.04 16.0 13 5.40 12 6.52 21.0 Sum of C7_Fat Version 2Sum of C7_Fat = 91 Sum of C7_Fit Sum of C7_Fit = 260 Mann-Whitney Test and CI: Fat, Fit N MedianFat 13 4.9300Fit 13 6.3800Point estimate for ETA1-ETA2 is -1.630095.4 Percent CI for ETA1-ETA2 is (-2.2200,-1.1901)W = 91.0Test of ETA1 = ETA2 vs ETA1 < ETA2 is significant at 0.0000The test is significant at 0.0000 (adjusted for ties) Paired T-Test and CI: Fat, Fit Paired T for Fat - Fit N Mean StDev SE MeanFat 13 4.671 0.708 0.196Fit 13 6.431 0.448 0.124Difference 13 -1.760 0.753 0.20995% upper bound for mean difference: -1.388T-Test of mean difference = 0 (vs < 0): T-Value = -8.43 P-Value = 0.000 Wilcoxon Signed Rank Test: diff Test of median = 0.000000 versus median < 0.000000 N for Wilcoxon Estimated N Test Statistic P Mediandiff 13 13 0.0 0.001 -1.765

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Data Display Table for Wilcoxon Signed Rank Test. Version 2 Sgnd Last two columns give Signed RankRow Fat Fit diff C11 rank C13 1 4.99 6.68 -1.69 1.69 7 -1 2 4.74 7.32 -2.58 2.58 11 -1 3 4.93 6.38 -1.45 1.45 6 -1 4 4.16 6.16 -2.00 2.00 8 -1 5 5.53 5.93 -0.40 0.40 1 -1 6 4.12 7.08 -2.96 2.96 13 -1 7 5.10 6.37 -1.27 1.27 4 -1 8 4.47 6.53 -2.06 2.06 9 -1 9 5.30 6.68 -1.38 1.38 5 -1 10 3.12 5.71 -2.59 2.59 12 -1 11 3.77 6.20 -2.43 2.43 10 -1 12 5.09 6.04 -0.95 0.95 2 -1 13 5.40 6.52 -1.12 1.12 3 -1 Data Display Table for Wilcoxon Signed Rank Test for Median of 2. Sgnd Last two columns give Signed RankRow diff C14 C15 C11 rank C13 1 -1.69 -2 0.31 0.31 3 1 2 -2.58 -2 -0.58 0.58 6 -1 3 -1.45 -2 0.55 0.55 5 1 4 -2.00 -2 0.00 0.00 1 0 5 -0.40 -2 1.60 1.60 13 1 6 -2.96 -2 -0.96 0.96 11 -1 7 -1.27 -2 0.73 0.73 9 1 8 -2.06 -2 -0.06 0.06 2 -1 9 -1.38 -2 0.62 0.62 8 1 10 -2.59 -2 -0.59 0.59 7 -1 11 -2.43 -2 -0.43 0.43 4 -1 12 -0.95 -2 1.05 1.05 12 1 13 -1.12 -2 0.88 0.88 10 1 Wilcoxon Signed Rank Test: C15 Test of median = 0.000000 versus median not = 0.000000 N for Wilcoxon Estimated N Test Statistic P MedianC15 13 12 53.0 0.290 0.2350 Wilcoxon Signed Rank Test: diff Test of median = -2.000 versus median not = -2.000 N for Wilcoxon Estimated N Test Statistic P Mediandiff 13 12 53.0 0.290 -1.765

Mean of C20 Mean of C20 = -1.76 Standard Deviation of C21 Standard deviation of C21 = 0.752695

Descriptive Statistics: C22 Variable N N* Mean SE Mean StDev Minimum Q1 Median Q3C22 13 0 -0.000 0.277 1.000 -1.594 -0.990 0.093 0.751


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Data Display Table for Lilliefors Test – Last Column is absolute D. Version 2Row dmoved dordered z_d Fe O cumO Fo Fe-Fo 1 -1.69 -2.96 -1.59427 0.055438 1 1 0.07692 0.021485 2 -2.58 -2.59 -1.10270 0.135078 1 2 0.15385 0.018768 3 -1.45 -2.58 -1.08942 0.137985 1 3 0.23077 0.092785 4 -2.00 -2.43 -0.89013 0.186697 1 4 0.30769 0.120995 5 -0.40 -2.06 -0.39857 0.345106 1 5 0.38462 0.039510 6 -2.96 -2.00 -0.31885 0.374919 1 6 0.46154 0.086620 7 -1.27 -1.69 0.09300 0.537048 1 7 0.53846 0.001414 8 -2.06 -1.45 0.41185 0.659777 1 8 0.61538 0.044392 9 -1.38 -1.38 0.50485 0.693169 1 9 0.69231 0.000861 10 -2.59 -1.27 0.65099 0.742475 1 10 0.76923 0.026756 11 -2.43 -1.12 0.85028 0.802415 1 11 0.84615 0.043739 12 -0.95 -0.95 1.07613 0.859066 1 12 0.92308 0.064011 13 -1.12 -0.40 1.80684 0.964606 1 13 1.00000 0.035394 Maximum of Fe-Fo In table above Fe column is Exact Cumulative Probability.Maximum of Fe-Fo = 0.120995Note that p-values for all three tests of Normality are above 5% - We cannot reject Normality.

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MTB > WOpen "C:\Documents and Settings\RBOVE\My Documents\Minitab\252x0821-02V3.MTW". Version 3 – Only data printouts are shown.Retrieving worksheet from file: 'C:\Documents and Settings\RBOVE\MyDocuments\Minitab\252x0821-02V3.MTW'Worksheet was saved on Wed Apr 02 2008Results for: 252x0821-02V3.MTWMTB > exec "252x0822.mtb"Executing from file: 252x0822.mtbDescriptive Statistics: Fat, Fit, diff Variable N N* Mean SE Mean StDev Minimum Q1 Median Q3Fat 13 0 4.632 0.199 0.718 3.120 4.140 4.930 5.200Fit 13 0 6.3615 0.0999 0.3602 5.7100 6.1000 6.3800 6.6050diff 13 0 -1.729 0.201 0.724 -2.960 -2.305 -1.690 -1.195

Variable MaximumFat 5.530Fit 7.0800diff -0.400

Covariances: Fat, Fit Diagonal Items are Variances. Fat FitFat 0.515186Fit 0.060379 0.129764 Data Display Row Fat fatsq Fit fitsq diff diffsq 1 4.99 24.9001 6.68 44.6224 -1.69 2.8561 2 4.24 17.9776 6.42 41.2164 -2.18 4.7524 3 4.93 24.3049 6.38 40.7044 -1.45 2.1025 4 4.16 17.3056 6.16 37.9456 -2.00 4.0000 5 5.53 30.5809 5.93 35.1649 -0.40 0.1600 6 4.12 16.9744 7.08 50.1264 -2.96 8.7616 7 5.10 26.0100 6.37 40.5769 -1.27 1.6129 8 4.47 19.9809 6.53 42.6409 -2.06 4.2436 9 5.30 28.0900 6.68 44.6224 -1.38 1.9044 10 3.12 9.7344 5.71 32.6041 -2.59 6.7081 11 3.77 14.2129 6.20 38.4400 -2.43 5.9049 12 5.09 25.9081 6.04 36.4816 -0.95 0.9025 13 5.40 29.1600 6.52 42.5104 -1.12 1.2544 Sum of Fat Sum of Fat = 60.22 Sum of Squares of Fat Sum of squares (uncorrected) of Fat = 285.140 Sum of fatsq Sum of fatsq = 285.140 Sum of Fit Sum of Fit = 82.7 Sum of Squares of Fit Sum of squares (uncorrected) of Fit = 527.656 Sum of fitsq Sum of fitsq = 527.656 Sum of diff Sum of diff = -22.48

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Sum of Squares of diff Version 3 Sum of squares (uncorrected) of diff = 45.1634 Sum of diffsq Sum of diffsq = 45.1634 Two-Sample T-Test and CI: Fat, Fit Two-sample T for Fat vs Fit N Mean StDev SE MeanFat 13 4.632 0.718 0.20Fit 13 6.362 0.360 0.10Difference = mu (Fat) - mu (Fit)Estimate for difference: -1.72995% upper bound for difference: -1.348T-Test of difference = 0 (vs <): T-Value = -7.76 P-Value = 0.000 DF = 24Both use Pooled StDev = 0.5679 Two-Sample T-Test and CI: Fat, Fit Two-sample T for Fat vs Fit N Mean StDev SE MeanFat 13 4.632 0.718 0.20Fit 13 6.362 0.360 0.10Difference = mu (Fat) - mu (Fit)Estimate for difference: -1.72995% upper bound for difference: -1.342T-Test of difference = 0 (vs <): T-Value = -7.76 P-Value = 0.000 DF = 17 Test for Equal Variances: Fat, Fit 95% Bonferroni confidence intervals for standard deviations N Lower StDev UpperFat 13 0.492089 0.717765 1.28355Fit 13 0.246967 0.360228 0.64418F-Test (Normal Distribution)Test statistic = 3.97, p-value = 0.024Levene's Test (Any Continuous Distribution)Test statistic = 4.17, p-value = 0.052Test for Equal Variances for Fat, Fit All graphs are at end of printout.


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Data Display Columns and Ranks for Wilcoxon-Mann-Whitney Test Version 3Row Fat C7_Fat Fit C7_Fit 1 4.99 8 6.68 24.5 2 4.24 5 6.42 21.0 3 4.93 7 6.38 20.0 4 4.16 4 6.16 17.0 5 5.53 13 5.93 15.0 6 4.12 3 7.08 26.0 7 5.10 10 6.37 19.0 8 4.47 6 6.53 23.0 9 5.30 11 6.68 24.5 10 3.12 1 5.71 14.0 11 3.77 2 6.20 18.0 12 5.09 9 6.04 16.0 13 5.40 12 6.52 22.0 Sum of C7_Fat Sum of C7_Fat = 91 Sum of C7_Fit Sum of C7_Fit = 260

Mann-Whitney Test and CI: Fat, Fit N MedianFat 13 4.9300Fit 13 6.3800Point estimate for ETA1-ETA2 is -1.590095.4 Percent CI for ETA1-ETA2 is (-2.2100,-1.1498)W = 91.0Test of ETA1 = ETA2 vs ETA1 < ETA2 is significant at 0.0000The test is significant at 0.0000 (adjusted for ties) Paired T-Test and CI: Fat, Fit Paired T for Fat - Fit N Mean StDev SE MeanFat 13 4.632 0.718 0.199Fit 13 6.362 0.360 0.100Difference 13 -1.729 0.724 0.20195% upper bound for mean difference: -1.371T-Test of mean difference = 0 (vs < 0): T-Value = -8.61 P-Value = 0.000 Wilcoxon Signed Rank Test: diff Test of median = 0.000000 versus median < 0.000000 N for Wilcoxon Estimated N Test Statistic P Mediandiff 13 13 0.0 0.001 -1.725

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Data Display Table for Wilcoxon Signed Rank Test. Version 3 Sgnd Last two columns give Signed Rank.Row Fat Fit diff C11 rank C13 1 4.99 6.68 -1.69 1.69 7 -1 2 4.24 6.42 -2.18 2.18 10 -1 3 4.93 6.38 -1.45 1.45 6 -1 4 4.16 6.16 -2.00 2.00 8 -1 5 5.53 5.93 -0.40 0.40 1 -1 6 4.12 7.08 -2.96 2.96 13 -1 7 5.10 6.37 -1.27 1.27 4 -1 8 4.47 6.53 -2.06 2.06 9 -1 9 5.30 6.68 -1.38 1.38 5 -1 10 3.12 5.71 -2.59 2.59 12 -1 11 3.77 6.20 -2.43 2.43 11 -1 12 5.09 6.04 -0.95 0.95 2 -1 13 5.40 6.52 -1.12 1.12 3 -1 Data Display Table for Wilcoxon Signed Rank Test for Median of 2. Sgnd Last two columns give Signed RankRow diff C14 C15 C11 rank C13 1 -1.69 -2 0.31 0.31 4 1 2 -2.18 -2 -0.18 0.18 3 -1 3 -1.45 -2 0.55 0.55 6 1 4 -2.00 -2 0.00 0.00 1 0 5 -0.40 -2 1.60 1.60 13 1 6 -2.96 -2 -0.96 0.96 11 -1 7 -1.27 -2 0.73 0.73 9 1 8 -2.06 -2 -0.06 0.06 2 -1 9 -1.38 -2 0.62 0.62 8 1 10 -2.59 -2 -0.59 0.59 7 -1 11 -2.43 -2 -0.43 0.43 5 -1 12 -0.95 -2 1.05 1.05 12 1 13 -1.12 -2 0.88 0.88 10 1 Wilcoxon Signed Rank Test: C15 Test of median = 0.000000 versus median not = 0.000000 N for Wilcoxon Estimated N Test Statistic P MedianC15 13 12 55.0 0.224 0.2750 Wilcoxon Signed Rank Test: diff Test of median = -2.000 versus median not = -2.000 N for Wilcoxon Estimated N Test Statistic P Mediandiff 13 12 55.0 0.224 -1.725 Mean of C20 Mean of C20 = -1.72923 Standard Deviation of C21 Standard deviation of C21 = 0.724010 Descriptive Statistics: C22 Variable N N* Mean SE Mean StDev Minimum Q1 Median Q3C22 13 0 -0.000 0.277 1.000 -1.700 -0.795 0.054 0.738


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Data Display Table for Lilliefors Test – Last Column is absolute D. Version 3Row dmoved dordered z_d Fe O cumO Fo Fe-Fo 1 -1.69 -2.96 -1.69993 0.044572 1 1 0.07692 0.032351 2 -2.18 -2.59 -1.18889 0.117241 1 2 0.15385 0.036605 3 -1.45 -2.43 -0.96790 0.166547 1 3 0.23077 0.064222 4 -2.00 -2.18 -0.62260 0.266774 1 4 0.30769 0.040919 5 -0.40 -2.06 -0.45686 0.323887 1 5 0.38462 0.060728 6 -2.96 -2.00 -0.37399 0.354208 1 6 0.46154 0.107331 7 -1.27 -1.69 0.05419 0.521606 1 7 0.53846 0.016855 8 -2.06 -1.45 0.38567 0.650130 1 8 0.61538 0.034746 9 -1.38 -1.38 0.48236 0.685223 1 9 0.69231 0.007084 10 -2.59 -1.27 0.63429 0.737053 1 10 0.76923 0.032177 11 -2.43 -1.12 0.84147 0.799957 1 11 0.84615 0.046197 12 -0.95 -0.95 1.07627 0.859097 1 12 0.92308 0.063980 13 -1.12 -0.40 1.83593 0.966816 1 13 1.00000 0.033184 Maximum of Fe-Fo Maximum of Fe-Fo = 0.107331Note that p-values for all three tests of Normality are above 5% - We cannot reject Normality.

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MTB > WOpen "C:\Documents and Settings\RBOVE\My Documents\Minitab\252x0821-02V4.MTW". Version 4 – Only data printouts are shown.Retrieving worksheet from file: 'C:\Documents and Settings\RBOVE\MyDocuments\Minitab\252x0821-02V4.MTW'Worksheet was saved on Wed Apr 02 2008Results for: 252x0821-02V4.MTWMTB > exec "252x0822.mtb"Executing from file: 252x0822.mtb

Descriptive Statistics: Fat, Fit, diff Variable N N* Mean SE Mean StDev Minimum Q1 Median Q3Fat 13 0 4.618 0.198 0.713 3.120 4.140 4.740 5.200Fit 13 0 6.434 0.124 0.448 5.710 6.100 6.420 6.680diff 13 0 -1.816 0.209 0.755 -2.960 -2.505 -2.000 -1.195

Variable MaximumFat 5.530Fit 7.320diff -0.400 Covariances: Fat, Fit Diagonal Items are Variances. Fat FitFat 0.508536Fit 0.069668 0.200626 Data Display Row Fat fatsq Fit fitsq diff diffsq 1 4.99 24.9001 6.68 44.6224 -1.69 2.8561 2 4.24 17.9776 6.42 41.2164 -2.18 4.7524 3 4.74 22.4676 7.32 53.5824 -2.58 6.6564 4 4.16 17.3056 6.16 37.9456 -2.00 4.0000 5 5.53 30.5809 5.93 35.1649 -0.40 0.1600 6 4.12 16.9744 7.08 50.1264 -2.96 8.7616 7 5.10 26.0100 6.37 40.5769 -1.27 1.6129 8 4.47 19.9809 6.53 42.6409 -2.06 4.2436 9 5.30 28.0900 6.68 44.6224 -1.38 1.9044 10 3.12 9.7344 5.71 32.6041 -2.59 6.7081 11 3.77 14.2129 6.20 38.4400 -2.43 5.9049 12 5.09 25.9081 6.04 36.4816 -0.95 0.9025 13 5.40 29.1600 6.52 42.5104 -1.12 1.2544 Sum of Fat Sum of Fat = 60.03 Sum of Squares of Fat Sum of squares (uncorrected) of Fat = 283.302 Sum of fatsq Sum of fatsq = 283.302 Sum of Fit Sum of Fit = 83.64 Sum of Squares of Fit Sum of squares (uncorrected) of Fit = 540.534 Sum of fitsq Sum of fitsq = 540.534

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Sum of diff Version 4Sum of diff = -23.61 Sum of Squares of diff Sum of squares (uncorrected) of diff = 49.7173 Sum of diffsq Sum of diffsq = 49.7173 Two-Sample T-Test and CI: Fat, Fit Two-sample T for Fat vs Fit N Mean StDev SE MeanFat 13 4.618 0.713 0.20Fit 13 6.434 0.448 0.12Difference = mu (Fat) - mu (Fit)Estimate for difference: -1.81695% upper bound for difference: -1.417T-Test of difference = 0 (vs <): T-Value = -7.78 P-Value = 0.000 DF = 24Both use Pooled StDev = 0.5955 Two-Sample T-Test and CI: Fat, Fit Two-sample T for Fat vs Fit N Mean StDev SE MeanFat 13 4.618 0.713 0.20Fit 13 6.434 0.448 0.12Difference = mu (Fat) - mu (Fit)Estimate for difference: -1.81695% upper bound for difference: -1.413T-Test of difference = 0 (vs <): T-Value = -7.78 P-Value = 0.000 DF = 20 Test for Equal Variances: Fat, Fit 95% Bonferroni confidence intervals for standard deviations N Lower StDev UpperFat 13 0.488903 0.713117 1.27524Fit 13 0.307082 0.447913 0.80099F-Test (Normal Distribution)Test statistic = 2.53, p-value = 0.121Levene's Test (Any Continuous Distribution)Test statistic = 3.17, p-value = 0.088Test for Equal Variances for Fat, Fit All graphs are at end of printout.


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Data Display Columns and Ranks for Wilcoxon-Mann-Whitney Test. Version 4Row Fat C7_Fat Fit C7_Fit 1 4.99 8 6.68 23.5 2 4.24 5 6.42 20.0 3 4.74 7 7.32 26.0 4 4.16 4 6.16 17.0 5 5.53 13 5.93 15.0 6 4.12 3 7.08 25.0 7 5.10 10 6.37 19.0 8 4.47 6 6.53 22.0 9 5.30 11 6.68 23.5 10 3.12 1 5.71 14.0 11 3.77 2 6.20 18.0 12 5.09 9 6.04 16.0 13 5.40 12 6.52 21.0 Sum of C7_Fat Sum of C7_Fat = 91 Sum of C7_Fit Sum of C7_Fit = 260 Mann-Whitney Test and CI: Fat, Fit N MedianFat 13 4.7400Fit 13 6.4200Point estimate for ETA1-ETA2 is -1.770095.4 Percent CI for ETA1-ETA2 is (-2.2899,-1.2301)W = 91.0Test of ETA1 = ETA2 vs ETA1 < ETA2 is significant at 0.0000The test is significant at 0.0000 (adjusted for ties) Paired T-Test and CI: Fat, Fit Paired T for Fat - Fit N Mean StDev SE MeanFat 13 4.618 0.713 0.198Fit 13 6.434 0.448 0.124Difference 13 -1.816 0.755 0.20995% upper bound for mean difference: -1.443T-Test of mean difference = 0 (vs < 0): T-Value = -8.67 P-Value = 0.000

Wilcoxon Signed Rank Test: diff Test of median = 0.000000 versus median < 0.000000 N for Wilcoxon Estimated N Test Statistic P Mediandiff 13 13 0.0 0.001 -1.850

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Data Display Table for Wilcoxon Signed Rank Test. Version 4 Sgnd Last two columns give Signed Rank.Row Fat Fit diff C11 rank C13 1 4.99 6.68 -1.69 1.69 6 -1 2 4.24 6.42 -2.18 2.18 9 -1 3 4.74 7.32 -2.58 2.58 11 -1 4 4.16 6.16 -2.00 2.00 7 -1 5 5.53 5.93 -0.40 0.40 1 -1 6 4.12 7.08 -2.96 2.96 13 -1 7 5.10 6.37 -1.27 1.27 4 -1 8 4.47 6.53 -2.06 2.06 8 -1 9 5.30 6.68 -1.38 1.38 5 -1 10 3.12 5.71 -2.59 2.59 12 -1 11 3.77 6.20 -2.43 2.43 10 -1 12 5.09 6.04 -0.95 0.95 2 -1 13 5.40 6.52 -1.12 1.12 3 -1 Data Display Table for Wilcoxon Signed Rank Test for Median of 2. Sgnd Last two columns give Signed Rank.Row diff C14 C15 C11 rank C13 1 -1.69 -2 0.31 0.31 4 1 2 -2.18 -2 -0.18 0.18 3 -1 3 -2.58 -2 -0.58 0.58 6 -1 4 -2.00 -2 0.00 0.00 1 0 5 -0.40 -2 1.60 1.60 13 1 6 -2.96 -2 -0.96 0.96 11 -1 7 -1.27 -2 0.73 0.73 9 1 8 -2.06 -2 -0.06 0.06 2 -1 9 -1.38 -2 0.62 0.62 8 1 10 -2.59 -2 -0.59 0.59 7 -1 11 -2.43 -2 -0.43 0.43 5 -1 12 -0.95 -2 1.05 1.05 12 1 13 -1.12 -2 0.88 0.88 10 1 Wilcoxon Signed Rank Test: C15 Test of median = 0.000000 versus median not = 0.000000 N for Wilcoxon Estimated N Test Statistic P MedianC15 13 12 50.0 0.410 0.1500 Wilcoxon Signed Rank Test: diff Test of median = -2.000 versus median not = -2.000 N for Wilcoxon Estimated N Test Statistic P Mediandiff 13 12 50.0 0.410 -1.850 Mean of C20 Mean of C20 = -1.81615 Standard Deviation of C21 Standard deviation of C21 = 0.754868 Descriptive Statistics: C22 Variable N N* Mean SE Mean StDev Minimum Q1 Median Q3C22 13 0 -0.000 0.277 1.000 -1.515 -0.913 -0.244 0.823


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Data Display Table for Lilliefors Test – Last Column is absolute D. Version 4Row dmoved dordered z_d Fe O cumO Fo Fe-Fo 1 -1.69 -2.96 -1.51529 0.064849 1 1 0.07692 0.012074 2 -2.18 -2.59 -1.02514 0.152648 1 2 0.15385 0.001198 3 -2.58 -2.58 -1.01189 0.155794 1 3 0.23077 0.074975 4 -2.00 -2.43 -0.81318 0.208056 1 4 0.30769 0.099636 5 -0.40 -2.18 -0.48200 0.314903 1 5 0.38462 0.069712 6 -2.96 -2.06 -0.32303 0.373336 1 6 0.46154 0.088203 7 -1.27 -2.00 -0.24355 0.403791 1 7 0.53846 0.134671 8 -2.06 -1.69 0.16712 0.566362 1 8 0.61538 0.049022 9 -1.38 -1.38 0.57779 0.718296 1 9 0.69231 0.025989 10 -2.59 -1.27 0.72351 0.765316 1 10 0.76923 0.003914 11 -2.43 -1.12 0.92222 0.821793 1 11 0.84615 0.024361 12 -0.95 -0.95 1.14742 0.874397 1 12 0.92308 0.048680 13 -1.12 -0.40 1.87603 0.969674 1 13 1.00000 0.030326

Maximum of Fe-Fo In table above Fe column is Exact Cumulative Probability.Maximum of Fe-Fo = 0.134671Note that p-values for all three tests of Normality are above 5% - We cannot reject Normality.

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MTB > WOpen "C:\Documents and Settings\RBOVE\My Documents\Minitab\252x0821-02V5.MTW". Version 5 – All commands and data printouts are shown.Retrieving worksheet from file: 'C:\Documents and Settings\RBOVE\MyDocuments\Minitab\252x0821-02V5.MTW'Worksheet was saved on Wed Apr 02 2008Results for: 252x0821-02V5.MTWMTB > exec "252x0822.mtb"Executing from file: 252x0822.mtb Descriptive Statistics: Fat, Fit, diff Variable N N* Mean SE Mean StDev Minimum Q1 Median Q3Fat 13 0 4.677 0.195 0.704 3.120 4.180 4.930 5.200Fit 13 0 6.451 0.122 0.441 5.710 6.120 6.420 6.680diff 13 0 -1.774 0.211 0.759 -2.960 -2.505 -1.690 -1.195

Variable MaximumFat 5.530Fit 7.320diff -0.400 Covariances: Fat, Fit Diagonal Items are Variances. Fat FitFat 0.495406Fit 0.056736 0.194308 Data Display Row Fat fatsq Fit fitsq diff diffsq 1 4.99 24.9001 6.68 44.6224 -1.69 2.8561 2 4.24 17.9776 6.42 41.2164 -2.18 4.7524 3 4.74 22.4676 7.32 53.5824 -2.58 6.6564 4 4.93 24.3049 6.38 40.7044 -1.45 2.1025 5 5.53 30.5809 5.93 35.1649 -0.40 0.1600 6 4.12 16.9744 7.08 50.1264 -2.96 8.7616 7 5.10 26.0100 6.37 40.5769 -1.27 1.6129 8 4.47 19.9809 6.53 42.6409 -2.06 4.2436 9 5.30 28.0900 6.68 44.6224 -1.38 1.9044 10 3.12 9.7344 5.71 32.6041 -2.59 6.7081 11 3.77 14.2129 6.20 38.4400 -2.43 5.9049 12 5.09 25.9081 6.04 36.4816 -0.95 0.9025 13 5.40 29.1600 6.52 42.5104 -1.12 1.2544 Sum of Fat Sum of Fat = 60.8 Sum of Squares of Fat Sum of squares (uncorrected) of Fat = 290.302 Sum of fatsq Sum of fatsq = 290.302 Sum of Fit Sum of Fit = 83.86 Sum of Squares of Fit Sum of squares (uncorrected) of Fit = 543.293


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Sum of diff Version 5Sum of diff = -23.06 Sum of Squares of diff Sum of squares (uncorrected) of diff = 47.8198 Sum of diffsq Sum of diffsq = 47.8198 Two-Sample T-Test and CI: Fat, Fit Two-sample T for Fat vs Fit N Mean StDev SE MeanFat 13 4.677 0.704 0.20Fit 13 6.451 0.441 0.12Difference = mu (Fat) - mu (Fit)Estimate for difference: -1.77495% upper bound for difference: -1.380T-Test of difference = 0 (vs <): T-Value = -7.70 P-Value = 0.000 DF = 24Both use Pooled StDev = 0.5872 Two-Sample T-Test and CI: Fat, Fit Two-sample T for Fat vs Fit N Mean StDev SE MeanFat 13 4.677 0.704 0.20Fit 13 6.451 0.441 0.12Difference = mu (Fat) - mu (Fit)Estimate for difference: -1.77495% upper bound for difference: -1.377T-Test of difference = 0 (vs <): T-Value = -7.70 P-Value = 0.000 DF = 20

Test for Equal Variances: Fat, Fit 95% Bonferroni confidence intervals for standard deviations N Lower StDev UpperFat 13 0.482550 0.703851 1.25867Fit 13 0.302208 0.440803 0.78827F-Test (Normal Distribution)Test statistic = 2.55, p-value = 0.119Levene's Test (Any Continuous Distribution)Test statistic = 1.74, p-value = 0.199Test for Equal Variances for Fat, Fit All graphs are at end of printout. Covariances: diff, diff Diagonal Items are Variances. diff diffdiff 0.576242diff 0.576242 0.576242 Probability Plot of diff Probability Plot of diff Probability Plot of diff

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Data Display Columns and Ranks for Wilcoxon-Mann-Whitney Test. Version 5Row Fat C7_Fat Fit C7_Fit 1 4.99 8 6.68 23.5 2 4.24 4 6.42 20.0 3 4.74 6 7.32 26.0 4 4.93 7 6.38 19.0 5 5.53 13 5.93 15.0 6 4.12 3 7.08 25.0 7 5.10 10 6.37 18.0 8 4.47 5 6.53 22.0 9 5.30 11 6.68 23.5 10 3.12 1 5.71 14.0 11 3.77 2 6.20 17.0 12 5.09 9 6.04 16.0 13 5.40 12 6.52 21.0

Sum of C7_Fat Sum of C7_Fat = 91 Sum of C7_Fit Sum of C7_Fit = 260 Mann-Whitney Test and CI: Fat, Fit N MedianFat 13 4.9300Fit 13 6.4200Point estimate for ETA1-ETA2 is -1.640095.4 Percent CI for ETA1-ETA2 is (-2.2300,-1.2300)W = 91.0Test of ETA1 = ETA2 vs ETA1 < ETA2 is significant at 0.0000The test is significant at 0.0000 (adjusted for ties) Paired T-Test and CI: Fat, Fit Paired T for Fat - Fit N Mean StDev SE MeanFat 13 4.677 0.704 0.195Fit 13 6.451 0.441 0.122Difference 13 -1.774 0.759 0.21195% upper bound for mean difference: -1.399T-Test of mean difference = 0 (vs < 0): T-Value = -8.43 P-Value = 0.000 Wilcoxon Signed Rank Test: diff Test of median = 0.000000 versus median < 0.000000 N for Wilcoxon Estimated N Test Statistic P Mediandiff 13 13 0.0 0.001 -1.775

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Data Display Table for Wilcoxon Signed Rank Test. Version 5 Sgnd Last two columns give Signed Rank.Row Fat Fit diff C11 rank C13 1 4.99 6.68 -1.69 1.69 7 -1 2 4.24 6.42 -2.18 2.18 9 -1 3 4.74 7.32 -2.58 2.58 11 -1 4 4.93 6.38 -1.45 1.45 6 -1 5 5.53 5.93 -0.40 0.40 1 -1 6 4.12 7.08 -2.96 2.96 13 -1 7 5.10 6.37 -1.27 1.27 4 -1 8 4.47 6.53 -2.06 2.06 8 -1 9 5.30 6.68 -1.38 1.38 5 -1 10 3.12 5.71 -2.59 2.59 12 -1 11 3.77 6.20 -2.43 2.43 10 -1 12 5.09 6.04 -0.95 0.95 2 -1 13 5.40 6.52 -1.12 1.12 3 -1 Data Display Table for Wilcoxon Signed Rank Test for Median of 2. Sgnd Last two columns give Signed Rank.Row diff C14 C15 C11 rank C13 1 -1.69 -2 0.31 0.31 3 1 2 -2.18 -2 -0.18 0.18 2 -1 3 -2.58 -2 -0.58 0.58 6 -1 4 -1.45 -2 0.55 0.55 5 1 5 -0.40 -2 1.60 1.60 13 1 6 -2.96 -2 -0.96 0.96 11 -1 7 -1.27 -2 0.73 0.73 9 1 8 -2.06 -2 -0.06 0.06 1 -1 9 -1.38 -2 0.62 0.62 8 1 10 -2.59 -2 -0.59 0.59 7 -1 11 -2.43 -2 -0.43 0.43 4 -1 12 -0.95 -2 1.05 1.05 12 1 13 -1.12 -2 0.88 0.88 10 1 Wilcoxon Signed Rank Test: C15 Test of median = 0.000000 versus median not = 0.000000 N for Wilcoxon Estimated N Test Statistic P MedianC15 13 13 60.0 0.328 0.2250 Wilcoxon Signed Rank Test: diff Test of median = -2.000 versus median not = -2.000 N for Wilcoxon Estimated N Test Statistic P Mediandiff 13 13 60.0 0.328 -1.775

Mean of C20 Mean of C20 = -1.77385 Standard Deviation of C21 Standard deviation of C21 = 0.759106 Descriptive Statistics: C22 Variable N N* Mean SE Mean StDev Minimum Q1 Median Q3C22 13 0 -0.000 0.277 1.000 -1.563 -0.963 0.110 0.763


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Data Display Table for Lilliefors Test – Last Column is absolute D. Version 5Row dmoved dordered z_d Fe O cumO Fo Fe-Fo 1 -1.69 -2.96 -1.56257 0.059077 1 1 0.07692 0.017846 2 -2.18 -2.59 -1.07515 0.141154 1 2 0.15385 0.012693 3 -2.58 -2.58 -1.06198 0.144123 1 3 0.23077 0.086646 4 -1.45 -2.43 -0.86438 0.193690 1 4 0.30769 0.114002 5 -0.40 -2.18 -0.53504 0.296310 1 5 0.38462 0.088305 6 -2.96 -2.06 -0.37696 0.353101 1 6 0.46154 0.108437 7 -1.27 -1.69 0.11045 0.543975 1 7 0.53846 0.005514 8 -2.06 -1.45 0.42662 0.665170 1 8 0.61538 0.049786 9 -1.38 -1.38 0.51883 0.698060 1 9 0.69231 0.005752 10 -2.59 -1.27 0.66374 0.746570 1 10 0.76923 0.022660 11 -2.43 -1.12 0.86134 0.805474 1 11 0.84615 0.040680 12 -0.95 -0.95 1.08528 0.861102 1 12 0.92308 0.061975 13 -1.12 -0.40 1.80982 0.964838 1 13 1.00000 0.035162 Maximum of Fe-Fo In table above Fe column is Exact Cumulative Probability.Maximum of Fe-Fo = 0.114002

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MTB > WOpen "C:\Documents and Settings\RBOVE\My Documents\Minitab\252x0821-02V6.MTW". Version 6 – All commands and data printouts are shown.Retrieving worksheet from file: 'C:\Documents and Settings\RBOVE\MyDocuments\Minitab\252x0821-02V6.MTW'Worksheet was saved on Wed Apr 02 2008Results for: 252x0821-02V6.MTWMTB > echoMTB > exec "252x0822.mtb"Executing from file: 252x0822.mtbMTB > #252x0822 Runs Fit-Fat problemMTB > # Change to descriptionMTB > describe c2 c3 c4Descriptive Statistics: Fat, Fit, diff Variable N N* Mean SE Mean StDev Minimum Q1 Median Q3Fat 13 0 4.572 0.185 0.667 3.120 4.140 4.740 5.095Fit 13 0 6.468 0.117 0.422 5.710 6.180 6.420 6.680diff 13 0 -1.897 0.177 0.638 -2.960 -2.505 -2.000 -1.325


MTB > covariance c2 c3Covariances: Fat, Fit Diagonal Items are Variances. Fat FitFat 0.444997Fit 0.108303 0.178414

MTB > erase c5-c100MTB > let c22 = c2*c2MTB > let c23 = c3*c3MTB > let c24 = c4*c4MTB > name c22 'fatsq'MTB > name c23 'fitsq'MTB > name c24 'diffsq'MTB > print c2 c22 c3 c23 c4 c24 Data Display Row Fat fatsq Fit fitsq diff diffsq 1 4.99 24.9001 6.68 44.6224 -1.69 2.8561 2 4.24 17.9776 6.42 41.2164 -2.18 4.7524 3 4.74 22.4676 7.32 53.5824 -2.58 6.6564 4 4.93 24.3049 6.38 40.7044 -1.45 2.1025 5 4.16 17.3056 6.16 37.9456 -2.00 4.0000 6 4.12 16.9744 7.08 50.1264 -2.96 8.7616 7 5.10 26.0100 6.37 40.5769 -1.27 1.6129 8 4.47 19.9809 6.53 42.6409 -2.06 4.2436 9 5.30 28.0900 6.68 44.6224 -1.38 1.9044 10 3.12 9.7344 5.71 32.6041 -2.59 6.7081 11 3.77 14.2129 6.20 38.4400 -2.43 5.9049 12 5.09 25.9081 6.04 36.4816 -0.95 0.9025 13 5.40 29.1600 6.52 42.5104 -1.12 1.2544

MTB > sum c2Sum of Fat Sum of Fat = 59.43

MTB > ssq c2Sum of Squares of Fat Sum of squares (uncorrected) of Fat = 277.027

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MTB > sum c22Sum of fatsq Version 6Sum of fatsq = 277.027

MTB > sum c3Sum of Fit Sum of Fit = 84.09

MTB > ssq c3Sum of Squares of Fit Sum of squares (uncorrected) of Fit = 546.074

MTB > sum c23Sum of fitsq Sum of fitsq = 546.074

MTB > sum c4Sum of diff Sum of diff = -24.66

MTB > ssq c4Sum of Squares of diff Sum of squares (uncorrected) of diff = 51.6598

MTB > sum c24Sum of diffsq Sum of diffsq = 51.6598

MTB > erase c22-c24MTB > twosample c2 c3;SUBC> pooled; #aSUBC> Alternative -1.Two-Sample T-Test and CI: Fat, Fit Two-sample T for Fat vs Fit N Mean StDev SE MeanFat 13 4.572 0.667 0.19Fit 13 6.468 0.422 0.12Difference = mu (Fat) - mu (Fit)Estimate for difference: -1.89795% upper bound for difference: -1.522T-Test of difference = 0 (vs <): T-Value = -8.66 P-Value = 0.000 DF = 24Both use Pooled StDev = 0.5583

MTB > twosample c2 c3; #bSUBC> Alternative -1.Two-Sample T-Test and CI: Fat, Fit Two-sample T for Fat vs Fit N Mean StDev SE MeanFat 13 4.572 0.667 0.19Fit 13 6.468 0.422 0.12Difference = mu (Fat) - mu (Fit)Estimate for difference: -1.89795% upper bound for difference: -1.519T-Test of difference = 0 (vs <): T-Value = -8.66 P-Value = 0.000 DF = 20

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MTB > VarTest c2 c3; #c Version 6SUBC> Unstacked.Test for Equal Variances: Fat, Fit 95% Bonferroni confidence intervals for standard deviations N Lower StDev UpperFat 13 0.457341 0.667081 1.19292Fit 13 0.289585 0.422391 0.75535F-Test (Normal Distribution)Test statistic = 2.49, p-value = 0.127Levene's Test (Any Continuous Distribution)Test statistic = 2.75, p-value = 0.111Test for Equal Variances for Fat, Fit All graphs are at end of printout.

MTB > covariance c4 c4 #dCovariances: diff, diff Diagonal Items are Variances. diff diffdiff 0.406806diff 0.406806 0.406806

MTB > NormTest c4;SUBC> KSTest.Probability Plot of diff MTB > NormTest c4.

Probability Plot of diff MTB > NormTest c4;SUBC> RJTest. Probability Plot of diff MTB > Stack c2 c3 c5 ; #eSUBC> Subscripts c6;SUBC> UseNames.MTB > Rank c5 c7.MTB > erase c8-c100MTB > Unstack (c7);SUBC> Subscripts c6;SUBC> After;SUBC> VarNames.MTB > print c2 c8 c3 c9Data Display Columns and Ranks for Wilcoxon-Mann-Whitney Test.Row Fat C7_Fat Fit C7_Fit 1 4.99 9 6.68 23.5 2 4.24 5 6.42 20.0 3 4.74 7 7.32 26.0 4 4.93 8 6.38 19.0 5 4.16 4 6.16 16.0 6 4.12 3 7.08 25.0 7 5.10 11 6.37 18.0 8 4.47 6 6.53 22.0 9 5.30 12 6.68 23.5 10 3.12 1 5.71 14.0 11 3.77 2 6.20 17.0 12 5.09 10 6.04 15.0 13 5.40 13 6.52 21.0

MTB > sum c8Sum of C7_Fat Sum of C7_Fat = 91

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MTB > sum c9 Version 6Sum of C7_Fit Sum of C7_Fit = 260

MTB > Mann-Whitney 95.0 c2 c3;SUBC> Alternative -1.Mann-Whitney Test and CI: Fat, Fit N MedianFat 13 4.7400Fit 13 6.4200

Point estimate for ETA1-ETA2 is -1.800095.4 Percent CI for ETA1-ETA2 is (-2.3298,-1.3802)W = 91.0Test of ETA1 = ETA2 vs ETA1 < ETA2 is significant at 0.0000The test is significant at 0.0000 (adjusted for ties)

MTB > Paired c2 c3; #fSUBC> Alternative -1.Paired T-Test and CI: Fat, Fit Paired T for Fat - Fit N Mean StDev SE MeanFat 13 4.572 0.667 0.185Fit 13 6.468 0.422 0.117Difference 13 -1.897 0.638 0.17795% upper bound for mean difference: -1.582T-Test of mean difference = 0 (vs < 0): T-Value = -10.72 P-Value = 0.000

MTB > WTest 0.0 c4; #gSUBC> Alternative -1.Wilcoxon Signed Rank Test: diff Test of median = 0.000000 versus median < 0.000000 N for Wilcoxon Estimated N Test Statistic P Mediandiff 13 13 0.0 0.001 -1.905

MTB > name c12 'sgnd rank'MTB > Rank c4 c10.MTB > let c11 = abs(c4)MTB > rank c11 c12MTB > let c13 = signs(c4)MTB > print c2 c3 c4 c11 c12 c13Data Display Sgnd Table for Wilcoxon Signed Rank Test.Row Fat Fit diff C11 rank C13 Last two columns give Signed Rank. 1 4.99 6.68 -1.69 1.69 6 -1 2 4.24 6.42 -2.18 2.18 9 -1 3 4.74 7.32 -2.58 2.58 11 -1 4 4.93 6.38 -1.45 1.45 5 -1 5 4.16 6.16 -2.00 2.00 7 -1 6 4.12 7.08 -2.96 2.96 13 -1 7 5.10 6.37 -1.27 1.27 3 -1 8 4.47 6.53 -2.06 2.06 8 -1 9 5.30 6.68 -1.38 1.38 4 -1 10 3.12 5.71 -2.59 2.59 12 -1 11 3.77 6.20 -2.43 2.43 10 -1 12 5.09 6.04 -0.95 0.95 1 -1 13 5.40 6.52 -1.12 1.12 2 -1

MTB > let c14 = 0 - 2*c2 #hMTB > let c14 = c14/c2

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MTB > let c15 = c4 + 2 Version 6MTB > let c11 = abs(c15)MTB > rank c11 c12MTB > let c13 = signs(c15)MTB > print c4 c14 c15 c11 c12 c13Data Display Table for Wilcoxon Signed Rank Test for Median of 2. Sgnd Last two columns give Signed Rank.Row diff C14 C15 C11 rank C13 1 -1.69 -2 0.31 0.31 4 1 2 -2.18 -2 -0.18 0.18 3 -1 3 -2.58 -2 -0.58 0.58 7 -1 4 -1.45 -2 0.55 0.55 6 1 5 -2.00 -2 0.00 0.00 1 0 6 -2.96 -2 -0.96 0.96 12 -1 7 -1.27 -2 0.73 0.73 10 1 8 -2.06 -2 -0.06 0.06 2 -1 9 -1.38 -2 0.62 0.62 9 1 10 -2.59 -2 -0.59 0.59 8 -1 11 -2.43 -2 -0.43 0.43 5 -1 12 -0.95 -2 1.05 1.05 13 1 13 -1.12 -2 0.88 0.88 11 1

MTB > WTest 0.0 c15Wilcoxon Signed Rank Test: C15 Test of median = 0.000000 versus median not = 0.000000 N for Wilcoxon Estimated N Test Statistic P MedianC15 13 12 47.0 0.556 0.09500

MTB > WTest -2.0 c4Wilcoxon Signed Rank Test: diff Test of median = -2.000 versus median not = -2.000 N for Wilcoxon Estimated N Test Statistic P Mediandiff 13 12 47.0 0.556 -1.905

MTB > erase c20 - c27 #d againMTB > let c20 = c4MTB > Sort c20 c21;SUBC> By c20.MTB > mean (c20)Mean of C20 Mean of C20 = -1.89692

MTB > let c22 = c21 - mean(c21)MTB > stdev c21Standard Deviation of C21 Standard deviation of C21 = 0.637814

MTB > let c22 = c22/stdev(c21)MTB > describe c22Descriptive Statistics: C22 Variable N N* Mean SE Mean StDev Minimum Q1 Median Q3C22 13 0 -0.000 0.277 1.000 -1.667 -0.953 -0.162 0.897


MTB > CDF c22 c23;SUBC> Normal 0.0 1.0.MTB > name C20 'dmoved'

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MTB > name c21 'dordered' Version 6MTB > name c22 'z_d'MTB > name c23 'Fe'MTB > Set c25DATA> 1( 1 : 13 / 1 )1DATA> End.MTB > name c25 'cumO'MTB > let c24 = c25/c25MTB > name c24 'O'MTB > name c25 'cumO'MTB > let c24 = c25/c25MTB > name C26 'Fo'MTB > let c26 = c25/count(c25)MTB > Name C27 'Fe-Fo'MTB > let c27 = c23 - c26MTB > let c27 = abs(c27)MTB > Print c20-c27Data Display Table for Lilliefors Test – Last Column is absolute D.Row dmoved dordered z_d Fe O cumO Fo Fe-Fo 1 -1.69 -2.96 -1.66675 0.047782 1 1 0.07692 0.029141 2 -2.18 -2.59 -1.08664 0.138597 1 2 0.15385 0.015249 3 -2.58 -2.58 -1.07097 0.142092 1 3 0.23077 0.088677 4 -1.45 -2.43 -0.83579 0.201637 1 4 0.30769 0.106055 5 -2.00 -2.18 -0.44382 0.328585 1 5 0.38462 0.056030 6 -2.96 -2.06 -0.25568 0.399099 1 6 0.46154 0.062440 7 -1.27 -2.00 -0.16161 0.435807 1 7 0.53846 0.102655 8 -2.06 -1.69 0.32443 0.627192 1 8 0.61538 0.011807 9 -1.38 -1.45 0.70071 0.758258 1 9 0.69231 0.065951 10 -2.59 -1.38 0.81046 0.791162 1 10 0.76923 0.021932 11 -2.43 -1.27 0.98292 0.837178 1 11 0.84615 0.008976 12 -0.95 -1.12 1.21810 0.888408 1 12 0.92308 0.034669 13 -1.12 -0.95 1.48464 0.931180 1 13 1.00000 0.068820

MTB > max (c27) In table above Fe column is Exact Cumulative Probability.Maximum of Fe-Fo Maximum of Fe-Fo = 0.106055

MTB > erase c20-c27MTB > end

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Note that p-values for all three tests of Normality are above 5% - We cannot reject Normality.

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MTB > WOpen "C:\Documents and Settings\RBOVE\My Documents\Minitab\252x0821-02V7.MTW". Version 7 – All commands and data printouts are shown.Retrieving worksheet from file: 'C:\Documents and Settings\RBOVE\MyDocuments\Minitab\252x0821-02V7.MTW'Worksheet was saved on Wed Apr 02 2008Results for: 252x0821-02V7.MTWMTB > exec "252x0822.mtb"Executing from file: 252x0822.mtbMTB > #252x0822 Runs Fit-Fat problemMTB > # Change to descriptionMTB > describe c2 c3 c4 Descriptive Statistics: Fat, Fit, diff Variable N N* Mean SE Mean StDev Minimum Q1 Median Q3Fat 13 0 4.680 0.195 0.701 3.120 4.200 4.930 5.200Fit 13 0 6.380 0.112 0.404 5.710 6.100 6.380 6.605diff 13 0 -1.700 0.188 0.676 -2.590 -2.305 -1.690 -1.195



MTB > erase c5-c100MTB > let c22 = c2*c2MTB > let c23 = c3*c3MTB > let c24 = c4*c4MTB > name c22 'fatsq'MTB > name c23 'fitsq'MTB > name c24 'diffsq'MTB > print c2 c22 c3 c23 c4 c24 Data Display Row Fat fatsq Fit fitsq diff diffsq 1 4.99 24.9001 6.68 44.6224 -1.69 2.8561 2 4.24 17.9776 6.42 41.2164 -2.18 4.7524 3 4.74 22.4676 7.32 53.5824 -2.58 6.6564 4 4.93 24.3049 6.38 40.7044 -1.45 2.1025 5 4.16 17.3056 6.16 37.9456 -2.00 4.0000 6 5.53 30.5809 5.93 35.1649 -0.40 0.1600 7 5.10 26.0100 6.37 40.5769 -1.27 1.6129 8 4.47 19.9809 6.53 42.6409 -2.06 4.2436 9 5.30 28.0900 6.68 44.6224 -1.38 1.9044 10 3.12 9.7344 5.71 32.6041 -2.59 6.7081 11 3.77 14.2129 6.20 38.4400 -2.43 5.9049 12 5.09 25.9081 6.04 36.4816 -0.95 0.9025 13 5.40 29.1600 6.52 42.5104 -1.12 1.2544



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MTB > sum c22 Version 7Sum of fatsq Sum of fatsq = 290.633









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MTB > VarTest c2 c3; #c Version 7UBC> Unstacked.Test for Equal Variances: Fat, Fit 95% Bonferroni confidence intervals for standard deviations N Lower StDev UpperFat 13 0.480799 0.701296 1.25410Fit 13 0.276737 0.403650 0.72183F-Test (Normal Distribution)Test statistic = 3.02, p-value = 0.067Levene's Test (Any Continuous Distribution)Test statistic = 2.37, p-value = 0.137Test for Equal Variances for Fat, Fit All graphs are at end of printout.


MTB > NormTest c4;SUBC> KSTest.Probability Plot of diff

MTB > NormTest c4.Probability Plot of diff

MTB > NormTest c4;SUBC> RJTest.Probability Plot of diff

MTB > Stack c2 c3 c5 ; #eSUBC> Subscripts c6;SUBC> UseNames.MTB > Rank c5 c7.MTB > erase c8-c100MTB > Unstack (c7);SUBC> Subscripts c6;SUBC> After;SUBC> VarNames.MTB > print c2 c8 c3 c9Data Display Columns and Ranks for Wilcoxon-Mann-Whitney Test.Row Fat C7_Fat Fit C7_Fit 1 4.99 8 6.68 24.5 2 4.24 4 6.42 21.0 3 4.74 6 7.32 26.0 4 4.93 7 6.38 20.0 5 4.16 3 6.16 17.0 6 5.53 13 5.93 15.0 7 5.10 10 6.37 19.0 8 4.47 5 6.53 23.0 9 5.30 11 6.68 24.5 10 3.12 1 5.71 14.0 11 3.77 2 6.20 18.0 12 5.09 9 6.04 16.0 13 5.40 12 6.52 22.0


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MTB > Mann-Whitney 95.0 c2 c3;SUBC> Alternative -1. Mann-Whitney Test and CI: Fat, Fit N MedianFat 13 4.9300Fit 13 6.3800Point estimate for ETA1-ETA2 is -1.580095.4 Percent CI for ETA1-ETA2 is (-2.1799,-1.1499)W = 91.0Test of ETA1 = ETA2 vs ETA1 < ETA2 is significant at 0.0000The test is significant at 0.0000 (adjusted for ties)



MTB > name c12 'sgnd rank'MTB > Rank c4 c10.MTB > let c11 = abs(c4)MTB > rank c11 c12MTB > let c13 = signs(c4)MTB > print c2 c3 c4 c11 c12 c13Data Display Table for Wilcoxon Signed Rank Test. Sgnd Last two columns give Signed Rank.Row Fat Fit diff C11 rank C13 1 4.99 6.68 -1.69 1.69 7 -1 2 4.24 6.42 -2.18 2.18 10 -1 3 4.74 7.32 -2.58 2.58 12 -1 4 4.93 6.38 -1.45 1.45 6 -1 5 4.16 6.16 -2.00 2.00 8 -1 6 5.53 5.93 -0.40 0.40 1 -1 7 5.10 6.37 -1.27 1.27 4 -1 8 4.47 6.53 -2.06 2.06 9 -1 9 5.30 6.68 -1.38 1.38 5 -1 10 3.12 5.71 -2.59 2.59 13 -1 11 3.77 6.20 -2.43 2.43 11 -1 12 5.09 6.04 -0.95 0.95 2 -1 13 5.40 6.52 -1.12 1.12 3 -1

MTB > let c14 = 0 - 2*c2 #hMTB > let c14 = c14/c2

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MTB > let c15 = c4 + 2 Version 7MTB > let c11 = abs(c15)MTB > rank c11 c12MTB > let c13 = signs(c15)MTB > print c4 c14 c15 c11 c12 c13 Data Display Table for Wilcoxon Signed Rank Test for Median of 2. Sgnd Last two columns give Signed Rank.Row diff C14 C15 C11 rank C13 1 -1.69 -2 0.31 0.31 4 1 2 -2.18 -2 -0.18 0.18 3 -1 3 -2.58 -2 -0.58 0.58 7 -1 4 -1.45 -2 0.55 0.55 6 1 5 -2.00 -2 0.00 0.00 1 0 6 -0.40 -2 1.60 1.60 13 1 7 -1.27 -2 0.73 0.73 10 1 8 -2.06 -2 -0.06 0.06 2 -1 9 -1.38 -2 0.62 0.62 9 1 10 -2.59 -2 -0.59 0.59 8 -1 11 -2.43 -2 -0.43 0.43 5 -1 12 -0.95 -2 1.05 1.05 12 1 13 -1.12 -2 0.88 0.88 11 1





MTB > let c22 = c22/stdev(c21)MTB > describe c22Descriptive Statistics: C22 Variable N N* Mean SE Mean StDev Minimum Q1 Median Q3C22 13 0 -0.000 0.277 1.000 -1.316 -0.895 0.015 0.747


MTB > CDF c22 c23;SUBC> Normal 0.0 1.0.

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MTB > name C20 'dmoved' Version 7MTB > name c21 'dordered'MTB > name c22 'z_d'MTB > name c23 'Fe'MTB > Set c25DATA> 1( 1 : 13 / 1 )1DATA> End.MTB > name c25 'cumO'MTB > let c24 = c25/c25MTB > name c24 'O'MTB > name c25 'cumO'MTB > let c24 = c25/c25MTB > name C26 'Fo'MTB > let c26 = c25/count(c25)MTB > Name C27 'Fe-Fo'MTB > let c27 = c23 - c26MTB > let c27 = abs(c27)MTB > Print c20-c27Data Display Table for Lilliefors Test – Last Column is absolute D.Row dmoved dordered z_d Fe O cumO Fo Fe-Fo 1 -1.69 -2.59 -1.31603 0.094082 1 1 0.07692 0.017159 2 -2.18 -2.58 -1.30124 0.096588 1 2 0.15385 0.057258 3 -2.58 -2.43 -1.07944 0.140196 1 3 0.23077 0.090573 4 -1.45 -2.18 -0.70977 0.238924 1 4 0.30769 0.068769 5 -2.00 -2.06 -0.53233 0.297250 1 5 0.38462 0.087365 6 -0.40 -2.00 -0.44361 0.328664 1 6 0.46154 0.132875 7 -1.27 -1.69 0.01479 0.505899 1 7 0.53846 0.032563 8 -2.06 -1.45 0.36967 0.644186 1 8 0.61538 0.028802 9 -1.38 -1.38 0.47318 0.681957 1 9 0.69231 0.010350 10 -2.59 -1.27 0.63583 0.737558 1 10 0.76923 0.031673 11 -2.43 -1.12 0.85764 0.804454 1 11 0.84615 0.041700 12 -0.95 -0.95 1.10901 0.866288 1 12 0.92308 0.056789 13 -1.12 -0.40 1.92229 0.972715 1 13 1.00000 0.027285



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MTB > WOpen "C:\Documents and Settings\RBOVE\My Documents\Minitab\252x0821-02V8.MTW". Version 8 – All commands and data printouts are shown.Retrieving worksheet from file: 'C:\Documents and Settings\RBOVE\MyDocuments\Minitab\252x0821-02V8.MTW'Worksheet was saved on Wed Apr 02 2008Results for: 252x0821-02V8.MTWMTB > exec "252x0822.mtb"Executing from file: 252x0822.mtbMTB > #252x0822 Runs Fit-Fat problemMTB > # Change to descriptionMTB > describe c2 c3 c4Descriptive Statistics: Fat, Fit, diff Variable N N* Mean SE Mean StDev Minimum Q1 Median Q3Fat 13 0 4.605 0.196 0.705 3.120 4.140 4.740 5.195Fit 13 0 6.435 0.124 0.448 5.710 6.100 6.420 6.680diff 13 0 -1.830 0.207 0.746 -2.960 -2.505 -2.000 -1.250



MTB > erase c5-c100MTB > let c22 = c2*c2MTB > let c23 = c3*c3MTB > let c24 = c4*c4MTB > name c22 'fatsq'MTB > name c23 'fitsq'MTB > name c24 'diffsq'MTB > print c2 c22 c3 c23 c4 c24Data Display Row Fat fatsq Fit fitsq diff diffsq 1 4.99 24.9001 6.68 44.6224 -1.69 2.8561 2 4.24 17.9776 6.42 41.2164 -2.18 4.7524 3 4.74 22.4676 7.32 53.5824 -2.58 6.6564 4 4.93 24.3049 6.38 40.7044 -1.45 2.1025 5 4.16 17.3056 6.16 37.9456 -2.00 4.0000 6 5.53 30.5809 5.93 35.1649 -0.40 0.1600 7 4.12 16.9744 7.08 50.1264 -2.96 8.7616 8 4.47 19.9809 6.53 42.6409 -2.06 4.2436 9 5.30 28.0900 6.68 44.6224 -1.38 1.9044 10 3.12 9.7344 5.71 32.6041 -2.59 6.7081 11 3.77 14.2129 6.20 38.4400 -2.43 5.9049 12 5.09 25.9081 6.04 36.4816 -0.95 0.9025 13 5.40 29.1600 6.52 42.5104 -1.12 1.2544



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MTB > erase c22-c24MTB > twosample c2 c3;SUBC> pooled; #aSUBC> Alternative -1. Two-Sample T-Test and CI: Fat, Fit Two-sample T for Fat vs Fit N Mean StDev SE MeanFat 13 4.605 0.705 0.20Fit 13 6.435 0.448 0.12Difference = mu (Fat) - mu (Fit)Estimate for difference: -1.83095% upper bound for difference: -1.434T-Test of difference = 0 (vs <): T-Value = -7.90 P-Value = 0.000 DF = 24Both use Pooled StDev = 0.5906


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MTB > Mann-Whitney 95.0 c2 c3;SUBC> Alternative -1.Mann-Whitney Test and CI: Fat, Fit N MedianFat 13 4.7400Fit 13 6.4200Point estimate for ETA1-ETA2 is -1.770095.4 Percent CI for ETA1-ETA2 is (-2.3000,-1.2399)W = 91.0Test of ETA1 = ETA2 vs ETA1 < ETA2 is significant at 0.0000The test is significant at 0.0000 (adjusted for ties)

MTB > Paired c2 c3; #fSUBC> Alternative -1.Paired T-Test and CI: Fat, Fit Paired T for Fat - Fit N Mean StDev SE MeanFat 13 4.605 0.705 0.196Fit 13 6.435 0.448 0.124Difference 13 -1.830 0.746 0.20795% upper bound for mean difference: -1.461T-Test of mean difference = 0 (vs < 0): T-Value = -8.85 P-Value = 0.000MTB > WTest 0.0 c4; #gSUBC> Alternative -1. Wilcoxon Signed Rank Test: diff Test of median = 0.000000 versus median < 0.000000 N for Wilcoxon Estimated N Test Statistic P Mediandiff 13 13 0.0 0.001 -1.850

MTB > name c12 'sgnd rank'MTB > Rank c4 c10.MTB > let c11 = abs(c4)MTB > rank c11 c12MTB > let c13 = signs(c4)MTB > print c2 c3 c4 c11 c12 c13Data Display Table for Wilcoxon Signed Rank Test. Sgnd Last two columns give Signed RankRow Fat Fit diff C11 rank C13 1 4.99 6.68 -1.69 1.69 6 -1 2 4.24 6.42 -2.18 2.18 9 -1 3 4.74 7.32 -2.58 2.58 11 -1 4 4.93 6.38 -1.45 1.45 5 -1 5 4.16 6.16 -2.00 2.00 7 -1 6 5.53 5.93 -0.40 0.40 1 -1 7 4.12 7.08 -2.96 2.96 13 -1 8 4.47 6.53 -2.06 2.06 8 -1 9 5.30 6.68 -1.38 1.38 4 -1 10 3.12 5.71 -2.59 2.59 12 -1 11 3.77 6.20 -2.43 2.43 10 -1 12 5.09 6.04 -0.95 0.95 2 -1 13 5.40 6.52 -1.12 1.12 3 -1

MTB > let c14 = 0 - 2*c2 #hMTB > let c14 = c14/c2MTB > let c15 = c4 + 2

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MTB > let c11 = abs(c15) Version 8MTB > rank c11 c12MTB > let c13 = signs(c15)MTB > print c4 c14 c15 c11 c12 c13Data Display Table for Wilcoxon Signed Rank Test for Median of 2. Sgnd Last two columns give Signed Rank.Row diff C14 C15 C11 rank C13 1 -1.69 -2 0.31 0.31 4 1 2 -2.18 -2 -0.18 0.18 3 -1 3 -2.58 -2 -0.58 0.58 7 -1 4 -1.45 -2 0.55 0.55 6 1 5 -2.00 -2 0.00 0.00 1 0 6 -0.40 -2 1.60 1.60 13 1 7 -2.96 -2 -0.96 0.96 11 -1 8 -2.06 -2 -0.06 0.06 2 -1 9 -1.38 -2 0.62 0.62 9 1 10 -2.59 -2 -0.59 0.59 8 -1 11 -2.43 -2 -0.43 0.43 5 -1 12 -0.95 -2 1.05 1.05 12 1 13 -1.12 -2 0.88 0.88 10 1







MTB > CDF c22 c23;SUBC> Normal 0.0 1.0.MTB > name C20 'dmoved'MTB > name c21 'dordered'

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MTB > name c22 'z_d' Version 8MTB > name c23 'Fe'MTB > Set c25DATA> 1( 1 : 13 / 1 )1DATA> End.MTB > name c25 'cumO'MTB > let c24 = c25/c25MTB > name c24 'O'MTB > name c25 'cumO'MTB > let c24 = c25/c25MTB > name C26 'Fo'MTB > let c26 = c25/count(c25)MTB > Name C27 'Fe-Fo'MTB > let c27 = c23 - c26MTB > let c27 = abs(c27)MTB > Print c20-c27Data Display Table for Lilliefors Test – Last Column is absolute D.Row dmoved dordered z_d Fe O cumO Fo Fe-Fo 1 -1.69 -2.96 -1.51554 0.064818 1 1 0.07692 0.012105 2 -2.18 -2.59 -1.01930 0.154030 1 2 0.15385 0.000184 3 -2.58 -2.58 -1.00589 0.157235 1 3 0.23077 0.073535 4 -1.45 -2.43 -0.80471 0.210493 1 4 0.30769 0.097199 5 -2.00 -2.18 -0.46941 0.319387 1 5 0.38462 0.065229 6 -0.40 -2.06 -0.30847 0.378861 1 6 0.46154 0.082677 7 -2.96 -2.00 -0.22800 0.409823 1 7 0.53846 0.128639 8 -2.06 -1.69 0.18777 0.574470 1 8 0.61538 0.040915 9 -1.38 -1.45 0.50965 0.694852 1 9 0.69231 0.002544 10 -2.59 -1.38 0.60353 0.726923 1 10 0.76923 0.042308 11 -2.43 -1.12 0.95224 0.829513 1 11 0.84615 0.016641 12 -0.95 -0.95 1.18024 0.881048 1 12 0.92308 0.042029 13 -1.12 -0.40 1.91789 0.972438 1 13 1.00000 0.027562

MTB > max (c27)Maximum of Fe-Fo In table above Fe column is Exact Cumulative Probability.Maximum of Fe-Fo = 0.128639


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MTB > WOpen "C:\Documents and Settings\RBOVE\My Documents\Minitab\252x0821-02V9.MTW". Version 9 – All commands and data printouts are shown.Retrieving worksheet from file: 'C:\Documents and Settings\RBOVE\MyDocuments\Minitab\252x0821-02V9.MTW'Worksheet was saved on Wed Apr 02 2008Results for: 252x0821-02V9.MTWMTB > exec "252x0822.mtb"Executing from file: 252x0822.mtbMTB > #252x0822 Runs Fit-Fat problemMTB > # Change to descriptionMTB > describe c2 c3 c4 Descriptive Statistics: Fat, Fit, diff Variable N N* Mean SE Mean StDev Minimum Q1 Median Q3Fat 13 0 4.653 0.199 0.717 3.120 4.140 4.930 5.200Fit 13 0 6.422 0.124 0.447 5.710 6.100 6.380 6.680diff 13 0 -1.769 0.210 0.757 -2.960 -2.505 -1.690 -1.195






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MTB > sum c4Sum of diff Sum of diff = -23





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MTB > Mann-Whitney 95.0 c2 c3;SUBC> Alternative -1.Mann-Whitney Test and CI: Fat, Fit N MedianFat 13 4.9300Fit 13 6.3800Point estimate for ETA1-ETA2 is -1.680095.4 Percent CI for ETA1-ETA2 is (-2.2602,-1.1700)W = 91.0Test of ETA1 = ETA2 vs ETA1 < ETA2 is significant at 0.0000The test is significant at 0.0000 (adjusted for ties)




MTB > let c14 = 0 - 2*c2 #hMTB > let c14 = c14/c2MTB > let c15 = c4 + 2

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MTB > let c11 = abs(c15) Version 9MTB > rank c11 c12MTB > let c13 = signs(c15)MTB > print c4 c14 c15 c11 c12 c13Data Display Table for Wilcoxon Signed Rank Test for Median of 2. Sgnd Last two columns give Signed Rank.Row diff C14 C15 C11 rank C13 1 -1.69 -2 0.31 0.31 3 1 2 -2.18 -2 -0.18 0.18 2 -1 3 -2.58 -2 -0.58 0.58 6 -1 4 -1.45 -2 0.55 0.55 5 1 5 -2.00 -2 0.00 0.00 1 0 6 -0.40 -2 1.60 1.60 13 1 7 -2.96 -2 -0.96 0.96 11 -1 8 -1.27 -2 0.73 0.73 9 1 9 -1.38 -2 0.62 0.62 8 1 10 -2.59 -2 -0.59 0.59 7 -1 11 -2.43 -2 -0.43 0.43 4 -1 12 -0.95 -2 1.05 1.05 12 1 13 -1.12 -2 0.88 0.88 10 1





MTB > let c22 = c22/stdev(c21)MTB > describe c22Descriptive Statistics: C22 Variable N N* Mean SE Mean StDev Minimum Q1 Median Q3C22 13 0 -0.000 0.277 1.000 -1.572 -0.971 0.105 0.758


MTB > CDF c22 c23;SUBC> Normal 0.0 1.0.MTB > name C20 'dmoved'MTB > name c21 'dordered'

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MTB > name c22 'z_d' Version 9

MTB > name c23 'Fe'MTB > Set c25DATA> 1( 1 : 13 / 1 )1DATA> End.MTB > name c25 'cumO'MTB > let c24 = c25/c25MTB > name c24 'O'MTB > name c25 'cumO'MTB > let c24 = c25/c25MTB > name C26 'Fo'MTB > let c26 = c25/count(c25)MTB > Name C27 'Fe-Fo'MTB > let c27 = c23 - c26MTB > let c27 = abs(c27)MTB > Print c20-c27Data Display Table for Lilliefors Test – Last Column is absolute D.Row dmoved dordered z_d Fe O cumO Fo Fe-Fo 1 -1.69 -2.96 -1.57218 0.057955 1 1 0.07692 0.018968 2 -2.18 -2.59 -1.08366 0.139257 1 2 0.15385 0.014589 3 -2.58 -2.58 -1.07046 0.142206 1 3 0.23077 0.088563 4 -1.45 -2.43 -0.87242 0.191491 1 4 0.30769 0.116201 5 -2.00 -2.18 -0.54234 0.293792 1 5 0.38462 0.090823 6 -0.40 -2.00 -0.30469 0.380303 1 6 0.46154 0.081236 7 -2.96 -1.69 0.10461 0.541657 1 7 0.53846 0.003195 8 -1.27 -1.45 0.42148 0.663298 1 8 0.61538 0.047914 9 -1.38 -1.38 0.51390 0.696340 1 9 0.69231 0.004032 10 -2.59 -1.27 0.65914 0.745096 1 10 0.76923 0.024135 11 -2.43 -1.12 0.85718 0.804328 1 11 0.84615 0.041826 12 -0.95 -0.95 1.08163 0.860292 1 12 0.92308 0.062785 13 -1.12 -0.40 1.80780 0.964681 1 13 1.00000 0.035319



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MTB > WOpen "C:\Documents and Settings\RBOVE\My Documents\Minitab\252x0821-02v10.MTW". Version 10 – All commands and data printouts are shown.Retrieving worksheet from file: 'C:\Documents and Settings\RBOVE\MyDocuments\Minitab\252x0821-02v10.MTW'Worksheet was saved on Wed Apr 02 2008Results for: 252x0821-02v10.MTWMTB > exec "252x0822.mtb"Executing from file: 252x0822.mtbMTB > #252x0822 Runs Fit-Fat problemMTB > # Change to descriptionMTB > describe c2 c3 c4Descriptive Statistics: Fat, Fit, diff Variable N N* Mean SE Mean StDev Minimum Q1 Median Q3Fat 13 0 4.589 0.192 0.691 3.120 4.140 4.740 5.095Fit 13 0 6.411 0.123 0.442 5.710 6.100 6.380 6.605diff 13 0 -1.822 0.208 0.752 -2.960 -2.505 -2.000 -1.195






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MTB > Stack c2 c3 c5 ; #eSUBC> Subscripts c6;SUBC> UseNames.MTB > Rank c5 c7.MTB > erase c8-c100MTB > Unstack (c7);SUBC> Subscripts c6;SUBC> After;SUBC> VarNames.MTB > print c2 c8 c3 c9Data Display Row Fat C7_Fat Fit C7_Fit Columns and Ranks for Wilcoxon-Mann-Whitney Test. 1 4.99 9 6.68 24 2 4.24 5 6.42 21 3 4.74 7 7.32 26 4 4.93 8 6.38 20 5 4.16 4 6.16 17 6 5.53 13 5.93 15 7 4.12 3 7.08 25 8 5.10 11 6.37 19 9 4.47 6 6.53 23 10 3.12 1 5.71 14 11 3.77 2 6.20 18 12 5.09 10 6.04 16 13 5.40 12 6.52 22


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MTB > Mann-Whitney 95.0 c2 c3;SUBC> Alternative -1.Mann-Whitney Test and CI: Fat, Fit N MedianFat 13 4.7400Fit 13 6.3800Point estimate for ETA1-ETA2 is -1.750095.4 Percent CI for ETA1-ETA2 is (-2.2699,-1.2796)W = 91.0Test of ETA1 = ETA2 vs ETA1 < ETA2 is significant at 0.0000




MTB > let c14 = 0 - 2*c2 #hMTB > let c14 = c14/c2MTB > let c15 = c4 + 2MTB > let c11 = abs(c15)

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MTB > rank c11 c12MTB > let c13 = signs(c15) Version 10MTB > print c4 c14 c15 c11 c12 c13Data Display Table for Wilcoxon Signed Rank Test for Median of 2. Sgnd Last two columns give Signed Rank.Row diff C14 C15 C11 rank C13 1 -1.69 -2 0.31 0.31 4 1 2 -2.18 -2 -0.18 0.18 3 -1 3 -2.58 -2 -0.58 0.58 7 -1 4 -1.45 -2 0.55 0.55 6 1 5 -2.00 -2 0.00 0.00 1 0 6 -0.40 -2 1.60 1.60 13 1 7 -2.96 -2 -0.96 0.96 11 -1 8 -1.27 -2 0.73 0.73 9 1 9 -2.06 -2 -0.06 0.06 2 -1 10 -2.59 -2 -0.59 0.59 8 -1 11 -2.43 -2 -0.43 0.43 5 -1 12 -0.95 -2 1.05 1.05 12 1 13 -1.12 -2 0.88 0.88 10 1




MTB > let c22 = c21 - mean(c21)TB > stdev c21Standard Deviation of C21 Standard deviation of C21 = 0.751741



MTB > CDF c22 c23;SUBC> Normal 0.0 1.0.MTB > name C20 'dmoved'MTB > name c21 'dordered'MTB > name c22 'z_d'

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MTB > name c23 'Fe' Version 10MTB > Set c25DATA> 1( 1 : 13 / 1 )1DATA> End.MTB > name c25 'cumO'MTB > let c24 = c25/c25MTB > name c24 'O'MTB > name c25 'cumO'MTB > let c24 = c25/c25MTB > name C26 'Fo'MTB > let c26 = c25/count(c25)MTB > Name C27 'Fe-Fo'MTB > let c27 = c23 - c26MTB > let c27 = abs(c27)MTB > Print c20-c27Data Display Table for Lilliefors Test – Last Column is absolute D.Row dmoved dordered z_d Fe O cumO Fo Fe-Fo 1 -1.69 -2.96 -1.51443 0.064958 1 1 0.07692 0.011965 2 -2.18 -2.59 -1.02224 0.153333 1 2 0.15385 0.000513 3 -2.58 -2.58 -1.00894 0.156502 1 3 0.23077 0.074268 4 -1.45 -2.43 -0.80940 0.209142 1 4 0.30769 0.098551 5 -2.00 -2.18 -0.47684 0.316737 1 5 0.38462 0.067878 6 -0.40 -2.06 -0.31721 0.375541 1 6 0.46154 0.085997 7 -2.96 -2.00 -0.23740 0.406174 1 7 0.53846 0.132287 8 -1.27 -1.69 0.17498 0.569452 1 8 0.61538 0.045933 9 -2.06 -1.45 0.49424 0.689431 1 9 0.69231 0.002877 10 -2.59 -1.27 0.73368 0.768429 1 10 0.76923 0.000802 11 -2.43 -1.12 0.93322 0.824646 1 11 0.84615 0.021507 12 -0.95 -0.95 1.15936 0.876845 1 12 0.92308 0.046232 13 -1.12 -0.40 1.89100 0.970688 1 13 1.00000 0.029312

MTB > max (c27)Maximum of Fe-Fo In table above Fe column is Exact Cumulative Probability.Maximum of Fe-Fo = 0.132287


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