3. [25 marks] Some female psychology students were investigating whether intelligence depends on brain size. They each took a standard test that measured verbal IQ and also underwent an MRI scan to measure their brain size. The resulting data is below.
IQ | BrainV | |
132 | 816.932 | |
132 | 951.545 | |
90 | 928.799 | |
136 | 991.305 | |
90 | 854.258 | |
129 | 833.868 | |
120 | 856.472 | |
100 | 878.897 | |
71 | 865.363 | |
132 | 852.244 | |
112 | 808.02 | |
129 | 790.619 | |
86 | 831.772 | |
90 | 798.612 | |
83 | 793.549 | |
126 | 866.662 | |
126 | 857.782 | |
90 | 834.344 | |
129 | 948.066 | |
86 | 893.983 |
The variables are:
IQ Result of the verbal IQ test. IQ is measured on an artificial index scale, usually meant to have an average value of 100.
BrainV Brain volume measured in thousands of ‘pixels’. The size of a pixel derives from the two-dimensional resolution of the MRI and the spacing of image ‘slices’. It can vary between scanners (and even between scans on the same scanner). The pixel size is the same for all scans in this study, but the standard volumetric equivalent (e.g., millilitres) is not known.
(c) As far as possible, assess the extent to which the model is appropriate to the data. [4]
(d) Leaving aside any problems identified in part (c), evaluate the evidence that verbal IQ is related to brain volume. Use a 5% significance level, and describe any relationship you find. [3]
IQ (Y) | BrainV (X) | X - Mx | Y - My | (X - Mx)2 | (X - Mx)(Y - My) |
132 | 816.932 | -45.7225 | 22.55 | 2090.547 | -1031.0424 |
132 | 951.545 | 88.8905 | 22.55 | 7901.521 | 2004.4808 |
90 | 928.799 | 66.1445 | -19.45 | 4375.0949 | -1286.5105 |
136 | 991.305 | 128.6505 | 26.55 | 16550.9512 | 3415.6708 |
90 | 854.258 | -8.3965 | -19.45 | 70.5012 | 163.3119 |
129 | 833.868 | -28.7865 | 19.55 | 828.6626 | -562.7761 |
120 | 856.47 | -6.1845 | 10.55 | 38.248 | -65.2465 |
100 | 878.90 | 16.2455 | -9.45 | 263.9163 | -153.52 |
71 | 865.36 | 2.7055 | -38.45 | 7.3197 | -104.0265 |
132 | 852.244 | -10.4105 | 22.55 | 108.3785 | -234.7568 |
112 | 808.02 | -54.6345 | 2.55 | 2984.9286 | -139.318 |
129 | 790.619 | -72.0355 | 19.55 | 5189.1133 | -1408.294 |
86 | 831.772 | -30.8825 | -23.45 | 953.7288 | 724.1946 |
90 | 798.612 | -64.0425 | -19.45 | 4101.4418 | 1245.6266 |
83 | 793.549 | -69.1055 | -26.45 | 4775.5701 | 1827.8405 |
126 | 866.662 | 4.0075 | 16.55 | 16.0601 | 66.3241 |
126 | 857.782 | -4.8725 | 16.55 | 23.7413 | -80.6399 |
90 | 834.344 | -28.3105 | -19.45 | 801.4844 | 550.6392 |
129 | 948.066 | 85.4115 | 19.55 | 7295.1243 | 1669.7948 |
86 | 893.983 | 31.3285 | -23.45 | 981.4749 | -734.6533 |
SS: 59357.8079 | SP: 5867.0995 |
Sum of X = 17253.09
Sum of Y = 2189
Mean X = 862.6545
Mean Y = 109.45
Sum of squares (SSX) = 59357.8079
Sum of products (SP) = 5867.0995
Regression Equation = ŷ = bX + a
b = SP/SSX = 5867.1/59357.81
= 0.09884
a = MY - bMX = 109.45 -
(0.1*862.65) = 24.1827
ŷ = 0.09884X + 24.1827
The value of R is 0.2549.
The value of R2, the coefficient of determination, is 0.065.
Excel Output
SUMMARY OUTPUT | ||||||||
Regression Statistics | ||||||||
Multiple R | 0.254932853 | |||||||
R Square | 0.06499076 | |||||||
Adjusted R Square | 0.013045802 | |||||||
Standard Error | 21.52910266 | |||||||
Observations | 20 | |||||||
ANOVA | ||||||||
df | SS | MS | F | Significance F | ||||
Regression | 1 | 579.9092977 | 579.9092977 | 1.25114664 | 0.278039633 | |||
Residual | 18 | 8343.040702 | 463.5022612 | |||||
Total | 19 | 8922.95 | ||||||
Coefficients | Standard Error | t Stat | P-value | Lower 95% | Upper 95% | Lower 95.0% | Upper 95.0% | |
Intercept | 24.18349953 | 76.38157836 | 0.316614294 | 0.75517875 | -136.2882417 | 184.6552407 | -136.2882417 | 184.6552407 |
BrainV (X) | 0.098841994 | 0.088366446 | 1.118546665 | 0.27803963 | -0.08680902 | 0.284493008 | -0.08680902 | 0.284493008 |
3. [25 marks] Some female psychology students were investigating whether intelligence depends on brain size. They each t...
3. [25 marks] Some female psychology students were investigating whether intelligence depends on brain size. They each took a standard test that measured verbal IQ and also underwent an MRI scan to measure their brain size. The resulting data is below, file named IQBrain.csv. IQ BrainV 132 816.932 132 951.545 90 928.799 136 991.305 90 854.258 129 833.868 120 856.472 100 878.897 71 865.363 132 852.244 112 808.02 129 790.619 86 831.772 90 798.612 83 793.549 126 866.662 126 857.782...
3. [25 marks] Some female psychology students were investigating whether intelligence depends on brain size. They each took a standard test that measured verbal IQ and also underwent an MRI scan to measure their brain size. The variables are: IQ Result of the verbal IQ test. IQ is measured on an artificial index scale, usually meant to have an average value of 100. BrainV Brain volume measured in thousands of ‘pixels’. The size of a pixel derives from the two-dimensional...
3. [25 marks] Some female psychology students were investigating whether intelligence depends on brain size. They each took a standard test that measured verbal IQ and also underwent an MRI scan to measure their brain size. The resulting data is below, file named IQBrain.csv. IQ BrainV 132 816.932 132 951.545 90 928.799 136 991.305 90 854.258 129 833.868 120 856.472 100 878.897 71 865.363 132 852.244 112 808.02 129 790.619 86 831.772 90 798.612 83 793.549 126 866.662 126 857.782...
Some female psychology students were investigating whether intelligence depends on brain size. They each took a standard test that measured verbal IQ and also underwent an MRI scan to measure their brain size. The resulting data are in the file IQBrain.csv. The variables are IQ Result of the verbal IQ test. IQ is measured on an artificial index scale, usually meant to have an average value of 100. BrainV Brain volume measured in thousands of ‘pixels’. The size of a...
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