A Researcher Randomly Samples Female (Sample 1) and Male Graduates of an MBA Program

252meanx5 10/13/06

Examples of tests for differences of means

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MTB > WOpen "C:\Documents and Settings\RBOVE\My Documents\Minitab\252-D.MTW". Retrieving worksheet from file: 'C:\Documents and Settings\RBOVE\My Documents\Minitab\252-D.MTW' Worksheet was saved on Fri Oct 13 2006

Results for: 252-D.MTW

MTB > Print c1 c2 Data Display The first data used for Method 1. This was done by method 3. Row D1_1 D1_2 1 305.47 344.53 2 306.48 344.53 3 324.22 344.53 4 334.75 341.47 5 320.61 305.77 6 299.40 298.63 7 299.40 292.51 8 293.34 305.77 9 322.63 389.41 10 294.35 322.09 11 319.60 328.21 12 287.28 361.87 13 290.31 319.03 14 302.43 337.39 15 263.04 291.49 16 307.48 350.65 17 323.64 341.47 18 304.45 369.01 19 310.51 348.61 20 312.53 338.41 21 288.29 309.85 22 307.48 330.25 23 291.32 370.03 24 319.60 327.19 25 328.69 298.63 26 284.25 367.99 27 310.51 293.53 28 314.55 338.41 29 300.41 332.29 30 309.50 341.47 31 303.44 338.41 32 332.73 334.33 33 329.70 329.23 34 321.62 321.07 35 293.34 336.37 36 324.65 348.61 37 322.63 318.01 38 317.58 310.87 39 311.52 319.03 40 311.52 301.69 41 337.78 342.49 42 357.98 344.53 43 291.32 340.45 44 272.13 307.81 45 326.67 333.31

1 252meanx5 10/13/06

46 305.46 326.17 47 277.18 303.73 48 304.45 354.73 49 304.45 342.49 50 320.61 310.87 51 313.54 323.11 52 295.36 339.43 53 315.56 350.65 54 282.23 353.71 55 304.45 303.73 56 270.11 320.05 57 301.42 329.23 58 308.49 315.97 59 331.72 332.29 60 293.34 356.77 61 252.94 312.91 62 316.57 325.15 63 340.81 310.87 64 297.38 324.13 65 307.48 324.13 66 304.45 329.23 67 290.31 348.61 68 304.45 320.05 69 293.34 342.49 70 285.26 346.57 71 303.44 369.01 72 333.74 330.25 73 323.64 305.77 74 280.21 364.93 75 288.29 341.47 76 270.11 321.07 77 264.05 327.19 78 302.43 333.31 79 293.34 331.27 80 274.15 316.99 81 272.13 314.95 82 334.75 336.37 83 277.18 303.73 84 318.59 345.55 85 284.25 318.01 86 303.44 322.09 87 303.44 331.27 88 273.14 336.37 89 273.14 332.29 90 286.27 324.13 91 352.93 318.01 92 274.15 323.11 93 286.27 329.23 94 311.52 327.19 95 292.33 335.35 96 262.03 303.73 97 284.25 313.93 98 311.52 369.01 99 276.17 335.35 100 286.27 319.03 101 331.72 311.89 102 311.52 310.87 103 331.72 346.57 104 290.31 348.61 105 288.29 322.09 106 317.58 334.33 107 315.56 347.59 108 320.61 338.41

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109 297.38 343.51 110 282.23 300.67 111 318.59 326.17 112 292.33 335.35 113 286.27 311.89 114 299.40 309.85 115 283.24 347.59 116 277.18 372.07 117 310.51 347.59 118 284.25 341.47 119 305.46 334.33 120 291.32 308.83 121 293.34 313.93 122 304.45 316.99 123 298.39 331.27 124 289.30 330.25 125 276.17 339.43 126 305.46 385.33 127 295.36 307.81 128 285.26 303.73 129 289.30 325.15 130 299.40 290.47 131 299.40 314.95 132 313.54 311.89 133 294.35 316.99 134 272.13 349.63 135 320.61 312.91 136 301.42 334.33 137 311.52 352.69 138 293.34 324.13 139 328.69 321.07 140 255.97 338.41 141 282.23 333.31 142 285.26 327.19 143 280.21 327.19 144 291.32 333.31 145 297.38 146 268.09 147 295.36 148 313.54 149 275.16 150 288.29 151 305.46 152 281.22 153 259.00 154 301.42 155 291.32 156 317.58 157 280.21 158 284.25 159 354.95 160 273.14 161 319.60 162 286.27 163 269.10 164 292.33 165 286.27 166 277.18 167 313.54 168 292.33 169 324.65

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MTB > describe c1 c2 Descriptive Statistics: D1_1, D1_2 Variable N N* Mean SE Mean StDev Minimum Q1 Median Q3 D1_1 169 0 300.00 1.54 20.00 252.94 286.27 299.40 313.54 D1_2 144 0 330.00 1.58 18.98 290.47 316.99 329.74 341.47

Variable Maximum D1_1 357.98 D1_2 389.41

MTB > TwoSample c1 c2.

Two-Sample T-Test and CI: D1_1, D1_2 Two-sample T for D1_1 vs D1_2

N Mean StDev SE Mean D1_1 169 300.0 20.0 1.5 D1_2 144 330.0 19.0 1.6

Difference = mu (D1_1) - mu (D1_2) Estimate for difference: -30.0069 95% CI for difference: (-34.3481, -25.6657) T-Test of difference = 0 (vs not =): T-Value = -13.60 P-Value = 0.000 DF = 307

MTB > print c3 c4 Data Display The data in the second example for method 1. Row D1_3 D1_4 1 15.752 21.132 2 19.680 20.281 3 20.662 25.458 4 17.716 18.581 5 21.644 20.281 6 15.752 17.728 7 16.734 22.834 8 20.662 25.387 9 16.734 20.281 10 17.716 20.281 11 19.680 20.281 12 24.590 13.473 13 21.644 22.834 14 17.716 27.940 15 13.788 19.430 16 16.734 21.983 17 17.716 23.685 18 21.644 18.579 19 15.752 20.281 20 24.590 28.791 21 17.716 21.983 22 14.770 10.920 23 11.824 20.281 24 17.716 16.026 25 24.590 28.791 26 23.608 21.132 27 18.698 24.536 28 18.698 19.430 29 18.698 22.834 30 19.680 27.940 31 21.644 25.387 32 13.788 25.387 33 14.770 13.473

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34 23.608 19.430 35 21.644 22.834 36 17.716 16.877 37 18.698 38 18.698 39 20.662 40 21.644 41 13.788 42 16.734 43 16.734 44 19.680 45 14.770 46 16.734 47 16.734 48 12.806 49 14.770 50 23.608 51 12.806 52 20.662 53 17.716 54 17.716 55 24.590 56 15.752 57 15.752 58 21.644 59 19.680 60 19.680 61 21.644 62 19.680 63 18.698 64 16.734 65 8.878 66 20.662 67 14.770 68 12.806 69 18.698 70 17.716 71 20.662 72 16.734 73 22.626 74 12.806 75 15.752 76 20.662 77 20.662 78 19.680 79 23.608 80 17.716 81 11.824 82 16.734 83 24.590 84 22.626 85 18.698 86 17.716 87 17.716 88 20.662 89 17.716 90 14.770 91 17.716 92 16.734 93 22.626 94 22.626 95 17.716 96 14.770

5 252meanx5 10/13/06

97 20.662 98 17.716 99 13.788 100 15.752 101 21.644 102 12.806 103 19.680 104 12.806 105 17.716 106 21.644 107 20.662 108 19.680 109 17.716 110 18.698 111 21.644 112 13.788 113 13.788 114 21.644 115 13.788 116 22.626 117 17.716 118 21.644 119 18.698 120 18.698 121 14.770

MTB > TwoSample c3 c4; SUBC> Alternative -1. Two-Sample T-Test and CI: D1_3, D1_4 Two-sample T for D1_3 vs D1_4

Difference = mu (D1_3) - mu (D1_4) Estimate for difference: -2.89917 95% upper bound for difference: -1.62189 T-Test of difference = 0 (vs <): T-Value = -3.81 P-Value = 0.000 DF = 48

MTB > print c5 c6

Data Display The data used for method 2.

Row D2_1 D2_2 1 53.32 45.02 2 58.44 74.04 3 49.32 116.14 4 53.24 103.76 5 25.60 67.90 6 61.34 85.44 7 41.56 60.88 8 60.49 67.90 9 44.47 71.41 10 21.19 63.52

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MTB > TwoSample c5 c6; SUBC> Pooled; SUBC> Confidence 99.0. Two-Sample T-Test and CI: D2_1, D2_2 The test is done with equal variances assumed Two-sample T for D2_1 vs D2_2

Difference = mu (D2_1) - mu (D2_2) Estimate for difference: -28.7040 99% CI for difference: (-51.6762, -5.7318) T-Test of difference = 0 (vs not =): T-Value = -3.60 P-Value = 0.002 DF = 18 Both use Pooled StDev = 17.8456

MTB > TwoSample c5 c6; SUBC> Confidence 99.0. Two-Sample T-Test and CI: D2_1, D2_2 The test is done with equal variances not assumed Two-sample T for D2_1 vs D2_2

Difference = mu (D2_1) - mu (D2_2) Estimate for difference: -28.7040 99% CI for difference: (-52.2211, -5.1869) T-Test of difference = 0 (vs not =): T-Value = -3.60 P-Value = 0.003 DF = 15

MTB > print c7 c8

Data Display The data used for method 3. Row D3_1 D3_2 1 5.67 10.37 2 8.08 7.16 3 8.31 3.65 4 7.63 6.98 5 7.01 6.84 6 8.24 4.51 7 4.84 4.55 8 7.14 9.82 9 10.41 7.21 10 9.16 8.64 11 6.98 6.72 12 10.72 9.10 13 11.88 9.08 14 5.82 4.16 15 8.27 7.87 16 11.04 6.94

MTB > describe c7 c8

Descriptive Statistics: D3_1, D3_2 Variable N N* Mean SE Mean StDev Minimum Q1 Median Q3 D3_1 16 0 8.200 0.506 2.025 4.840 6.988 8.160 10.098 D3_2 16 0 7.100 0.512 2.049 3.650 5.093 7.070 8.970

Variable Maximum D3_1 11.880 D3_2 10.370

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MTB > TwoSample c7 c8. Two-Sample T-Test and CI: D3_1, D3_2 Not assuming equal variances Two-sample T for D3_1 vs D3_2

Difference = mu (D3_1) - mu (D3_2) Estimate for difference: 1.10000 95% CI for difference: (-0.37310, 2.57310) T-Test of difference = 0 (vs not =): T-Value = 1.53 P-Value = 0.138 DF = 29

MTB > TwoSample c7 c8; SUBC> Pooled. Two-Sample T-Test and CI: D3_1, D3_2 Assuming equal variances Two-sample T for D3_1 vs D3_2

Difference = mu (D3_1) - mu (D3_2) Estimate for difference: 1.10000 95% CI for difference: (-0.37097, 2.57097) T-Test of difference = 0 (vs not =): T-Value = 1.53 P-Value = 0.137 DF = 30 Both use Pooled StDev = 2.0372

MTB > print c9 - c11 Data Display One version of the data used for method 4. Row D4_1 D4_2 D4-D1 1 4110.4 4117.4 -7.00 2 6692.4 6694.1 -1.76 3 13104.6 13110.3 -5.73 4 4378.3 4383.0 -4.74 5 7686.9 7689.2 -2.35 6 6221.8 6222.7 -0.97 7 5143.3 5142.2 1.17 8 12332.8 12351.0 -18.15 9 9153.2 9156.0 -2.81 10 2403.2 2405.8 -2.60 11 9279.9 9274.3 5.61 12 9283.9 9290.7 -6.81 13 23797.9 23806.3 -8.35 14 8036.2 8046.8 -10.56 15 10401.6 10397.1 4.45 16 9996.9 9994.3 2.67 17 5160.3 5170.3 -10.03 18 5849.7 5841.4 8.25 19 11086.0 11086.9 -0.90 20 6484.2 6495.0 -10.84 21 11694.8 11692.4 2.38 22 7558.8 7563.6 -4.78 23 11460.4 11469.7 -9.28 24 10342.2 10353.8 -11.60 25 6333.0 6337.2 -4.15 26 8465.1 8475.3 -10.25 27 11383.0 11388.1 -5.07 28 3304.7 3295.3 9.48 29 2445.3 2455.0 -9.71

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30 6558.6 6566.7 -8.12 31 9167.6 9172.5 -4.91 32 13339.6 13342.2 -2.56 33 9341.7 9343.0 -1.32 34 12766.1 12777.4 -11.32 35 17813.9 17816.5 -2.61 36 6532.8 6539.5 -6.77 37 14936.0 14934.8 1.28 38 18102.2 18104.9 -2.74 39 8819.4 8820.3 -0.84 40 9293.3 9304.2 -10.97 41 5007.7 5014.9 -7.21 42 4834.8 4834.3 0.49 43 7869.2 7869.6 -0.42 44 10013.0 10025.0 -11.98 45 6596.8 6605.3 -8.54 46 7629.4 7635.4 -6.01 47 11280.1 11281.0 -0.98 48 7191.2 7196.3 -5.14 49 7408.6 7416.5 -7.85 50 15711.9 15721.9 -10.01 51 12233.8 12237.1 -3.35 52 8955.3 8947.0 8.30 53 2916.1 2921.6 -5.55 54 7354.9 7358.1 -3.29 55 16215.5 16223.3 -7.74 56 4381.1 4383.2 -2.14 57 10471.0 10483.0 -12.01 58 18366.7 18377.7 -11.01 59 8910.3 8911.8 -1.41 60 8872.6 8878.6 -6.00 61 7954.8 7963.7 -8.87 62 18437.2 18443.5 -6.29 63 5311.6 5319.1 -7.47 64 4688.1 4685.9 2.11 65 3649.2 3659.3 -10.14 66 7488.0 7497.5 -9.41 67 7706.9 7716.2 -9.31 68 5817.1 5821.2 -4.09 69 14741.5 14745.8 -4.37 70 6199.4 6210.5 -11.03 71 6418.6 6430.9 -12.32 72 6458.2 6473.9 -15.71 73 23574.2 23570.1 4.10 74 10627.7 10619.0 8.65 75 8576.6 8576.6 0.02 76 9501.9 9504.8 -2.90 77 8444.0 8452.0 -8.05 78 5861.1 5867.4 -6.26 79 19260.6 19268.0 -7.39 80 5048.6 5043.5 5.10 81 9178.5 9180.1 -1.64 82 11334.1 11332.8 1.35 83 7451.1 7452.0 -0.84 84 7935.2 7951.9 -16.74 85 3177.6 3176.7 0.83 86 7526.2 7525.0 1.22 87 3835.6 3842.2 -6.62 88 5054.1 5051.0 3.10 89 14424.9 14419.6 5.39 90 13181.5 13190.9 -9.37 91 8633.3 8633.8 -0.54 92 11906.2 11896.4 9.75

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93 11575.6 11577.4 -1.81 94 8401.1 8399.6 1.55 95 1371.3 1370.5 0.80 96 7463.0 7466.1 -3.08 97 9644.3 9634.7 9.61 98 10033.5 10041.7 -8.21 99 7506.8 7517.9 -11.06 100 9716.0 9718.9 -2.94

MTB > Paired 'D4_1' 'D4_2'; SUBC> Test 4. Paired T-Test and CI: D4_1, D4_2 Paired T for D4_1 - D4_2

N Mean StDev SE Mean D4_1 100 9075.97 4315.72 431.57 D4_2 100 9079.97 4315.86 431.59 Difference 100 -4.00040 5.99961 0.59996

95% CI for mean difference: (-5.19085, -2.80995) T-Test of mean difference = 4 (vs not = 4): T-Value = -13.33 P-Value = 0.000

MTB > print c12-c14 Data Display A more realistic version of the Method 4 example Row D4_3 D4_4 D4_D2 1 4110 4117 -7 2 6692 6694 -2 3 13105 13110 -5 4 4378 4383 -5 5 7687 7689 -2 6 6222 6223 -1 7 5143 5142 1 8 12333 12350 -17 9 9153 9156 -3 10 2403 2406 -3 11 9280 9274 6 12 9284 9291 -7 13 23798 23806 -8 14 8036 8047 -11 15 10402 10397 5 16 9997 9994 3 17 5160 5169 -9 18 5850 5841 9 19 11086 11088 -2 20 6484 6493 -9 21 11695 11692 3 22 7559 7565 -6 23 11460 11472 -12 24 10342 10354 -12 25 6333 6338 -5 26 8465 8475 -10 27 11383 11388 -5 28 3305 3295 10 29 2445 2455 -10 30 6559 6567 -8 31 9168 9173 -5 32 13340 13342 -2 33 9342 9343 -1 34 12766 12777 -11 35 17814 17816 -2 36 6533 6540 -7 37 14936 14935 1 38 18102 18105 -3

10 252meanx5 10/13/06

39 8819 8820 -1 40 9293 9304 -11 41 5008 5015 -7 42 4835 4834 1 43 7869 7870 -1 44 10013 10025 -12 45 6597 6605 -8 46 7629 7635 -6 47 11280 11281 -1 48 7191 7196 -5 49 7409 7416 -7 50 15712 15722 -10 51 12234 12237 -3 52 8955 8947 8 53 2916 2922 -6 54 7355 7358 -3 55 16216 16223 -7 56 4381 4383 -2 57 10471 10483 -12 58 18367 18378 -11 59 8910 8912 -2 60 8873 8879 -6 61 7955 7964 -9 62 18437 18443 -6 63 5312 5319 -7 64 4688 4686 2 65 3649 3659 -10 66 7488 7497 -9 67 7707 7716 -9 68 5817 5822 -5 69 14741 14746 -5 70 6199 6210 -11 71 6419 6431 -12 72 6458 6474 -16 73 23574 23570 4 74 10628 10619 9 75 8577 8578 -1 76 9502 9506 -4 77 8444 8452 -8 78 5861 5867 -6 79 19261 19268 -7 80 5049 5043 6 81 9178 9180 -2 82 11334 11333 1 83 7451 7452 -1 84 7935 7952 -17 85 3178 3177 1 86 7526 7525 1 87 3836 3842 -6 88 5054 5051 3 89 14425 14420 5 90 13182 13191 -9 91 8633 8634 -1 92 11906 11896 10 93 11576 11577 -1 94 8401 8400 1 95 1371 1370 1 96 7463 7466 -3 97 9644 9635 9 98 10033 10042 -9 99 7507 7518 -11 100 9716 9720 -4

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MTB > Paired 'D4_3' 'D4_4'; SUBC> Test 4; SUBC> GHistogram; SUBC> GIndPlot; SUBC> GBoxplot. Paired T-Test and CI: D4_3, D4_4 Paired T for D4_3 - D4_4 N Mean StDev SE Mean D4_3 100 9075.98 4315.75 431.57 D4_4 100 9079.98 4315.87 431.59 Difference 100 -4.00000 5.99495 0.59949

95% CI for mean difference: (-5.18953, -2.81047) T-Test of mean difference = 4 (vs not = 4): T-Value = -13.34 P-Value = 0.000

Histogram of Differences This is shown below Individual Value Plot of Differences Boxplot of Differences

MTB > NormTest 'D4_3'; Note that the graphs below show a definite SUBC> Title "Before". Skewness in both the before and after data Before For the paired (Method 4) example. The graph of the difference between them is somewhat MTB > NormTest 'D4_3'; less skewed and gives a higher p-value for the SUBC> Title "After". null hypothesis of Normality. After

MTB > NormTest 'D4_D2'; SUBC> Title "Difference". Difference