1 answer

3. [25 marks] Some female psychology students were investigating whether intelligence depends on brain size. They each t...

Question:

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.

Whether intelligence depends on brain size 130 o 120 110 E 100 90 80 70 800 850 900 950 BrainV (ml)

Whether intelligence depends on brain size (0D 2 70 80 90 100 110 120 130 BrainV (ml)

Distribution of Standardised Residuals 2 2.0 1.5 1.0-0.5 0.0 0.5 1.0 1.5 Standardised residual

(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]

Whether intelligence depends on brain size 130 o 120 110 E 100 90 80 70 800 850 900 950 BrainV (ml)
Whether intelligence depends on brain size (0D 2 70 80 90 100 110 120 130 BrainV (ml)
Distribution of Standardised Residuals 2 2.0 1.5 1.0-0.5 0.0 0.5 1.0 1.5 Standardised residual

Answers

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.

5 X Values

Whether intelligence depends on brain size 130o 120 - 110 100 ー90-1 80 70 3 800 850 900 950 BrainV (ml)

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

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