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Question 3 [22 marks] Some scientists argue that mathematical ability is a heritable trait

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Question 3 [22 marks]

Some scientists argue that mathematical ability is a heritable trait. A study was
conducted to determine how a person's IQ(Y) may be affected by their parent's average
1Q (X,) and their parent's average university GPA (X2). The following regression model
was fitted to a random sample of 50 people:

Y = Bot WX, + Xr +. Q)

Some regression output is provided below (which is repeated on subsequent pages for

convenience) and we are also given the following sample statistics: 5} = 134.6208, s%, =

118.0373, 6%, = 0.479561, sx,y = 120.0287, sx.¥ = 5.616062 and sx,x, = 5.684412,

Predictor Coef SE Coef T p-value
Intercept -1.7805 4.8277 -0.37 0.7139
x, 1.0553 0.0723 «14.60 0.0000
Xe 0.7981 1.1339 -0.70 _ 0.4850

(a). (4 points) The standard error of estimate for the multiple regression model was
Ss, = 3.600712. Teast the overall significance of the model. Clearly state your hypotheses
and use a significance level of a = 5%.

(b). (2 points) Calculate the sample correlation coefficient between IQ(Y) and parent’s
average university GPA (X2).

(c). (3 points) What do you notice regarding the parameter estimates and your answer
to part (b)? Give an explanation as to why this might be occurring.

(d). (3 points) The scientist conducting this study believes thet the coefficient parameter
for X2 should be smaller than the coefficient parameter for X,. Test whether 52 <
1.0553. Clearly state your hypotheses and use a significance level of a = 5%.

The scientist wants more analysis done before making any conclusions regarding 8,
and 3). He collected 6 more independent random samples, each of size 50, and fitted the 

regression model Y = 9 + 9X) + 2:.X2 + € to each sample. The parameter estimates
obtained from each sample are summarised in the table below:

Sample 1 2 3 4 5 6

Bo -2.90 1.04 0.85 -7.30 -0.16 3.77

A 1.03 0.98 1.07 1.10 1.01 0.95

b 0.10 0.36 -1.95 -0.69 -0.45 0.51
(e) (5 points) Based on this data, test whether 2 is less than 1. Clearly state your

hypotheses and use a significance level of a = 5%.

The scientist decides to focus on how a person’s IQ(Y) is related to their parent’s
average 1Q(X;) and their handedness. Assuming that people can actually be classified as
left-handed, right-handed or ambidextrous (adept with both hands), the scientist wants
to answer the following

@ Is there a linear relationship between a person's IQ and their parent’s average 1Q?

@ Is a different intercept needed for left-handed people?

@ Is a different intercept needed for ambidextrous people?

@ Does the coefficient for parent’s average IQ differ for left-handed people?

@ Does the coefficient for parent’s average IQ differ for ambidextrous people?

(f} (4 points} Write down the model that the scientist needs to fit in order to answer all
the above questions. Clearly define all variables in your model.
(g} (1 point) Based on your model in part (f), what form does the model take for right-
handed people?