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, 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
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.

(a) Obtain the following R output, treating IQ as the response variable:

(i) scatter plot [3]

(ii) standardised residual plot [2]

(iii) distribution of residuals

I also found this info using R studio:

Call:

lm(formula = BrainV ~ IQ)

Residuals:

   Min     1Q Median     3Q    Max

-84.89 -44.04 -14.29 32.68 111.19

Coefficients:

            Estimate Std. Error t value Pr(>|t|)   

(Intercept) 790.6888    65.5258 12.067 4.61e-10 ***

IQ            0.6575     0.5878   1.119    0.278   

---

Signif. codes: 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1

Residual standard error: 55.53 on 18 degrees of freedom

Multiple R-squared: 0.06499, Adjusted R-squared: 0.01305

F-statistic: 1.251 on 1 and 18 DF, p-value: 0.278

I have done Q1) i, ii, iii and iv but need help with rest of the questions. Thanks

(iv) regression analysis (summary of linear model). [2, total = 10]

Call:

lm(formula = BrainV ~ IQ)

Residuals:

   Min     1Q Median     3Q    Max

-84.89 -44.04 -14.29 32.68 111.19

Coefficients:

            Estimate Std. Error t value Pr(>|t|)   

(Intercept) 790.6888    65.5258 12.067 4.61e-10 ***

IQ            0.6575     0.5878   1.119    0.278   

---

Signif. codes: 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1

Residual standard error: 55.53 on 18 degrees of freedom

Multiple R-squared: 0.06499, Adjusted R-squared: 0.01305

F-statistic: 1.251 on 1 and 18 DF, p-value: 0.278

(b) What statistical model is assumed in your regression analysis? [2]

(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]

(e) Provide a 95% prediction interval for the verbal IQ of a female whose brain volume was 870,000 pixels. [4]

(f) One of the students showed the scatterplot to a friend who said, “Well, there is clearly a trend there. It looks like people with larger brains do have higher IQ.” Write a brief response to this comment. [2]

Whether intelligence depends on brain size 950 900 850 800 130 120 100 110 90 30) 70 IC.) Whether intelligence depends on brain size (0D 2 70 80 90 100 110 120 130 IQ (index scale) Distribution of Standardised Residuals 2 -2 -1 Standardised residual

Answer 1

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