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.

(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

Answer 1

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 (SS_X) = 59357.8079
Sum of products (SP) = 5867.0995

Regression Equation = ŷ = bX + a

b = SP/SS_X = 5867.1/59357.81 = 0.09884

a = M_Y - bM_X = 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

Excel Output

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

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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