Define the term actuary

Q4

An actuary is asked to check a linear regression calculation performed by a trainee.

The trainee reports a least squares slope parameter estimate of = 13.7 and a sample

correlation coefficient = 0.89.

(i) Justify why this suggests that the trainee has made an error. [2]

In a different simple linear regression model, a histogram of the residuals is shown

below.

(ii) Comment on the validity of the assumptions of the linear model. [2]

The following pairs of data are available:

x: 0 1 2 3 4 5 6 7 8 9

y: 1.35 4.96 9.20 13.15 16.70 21.23 25.14 28.44 33.68 37.39

for which

= 19.124, ( )2 = 1,329.523,

10

=1

( )2 = 82.5

10

=1

( )( ) = 331.05

10

=1

A linear model of the form y = a + bx + e is fitted to the data, where the error terms

(e) independently follow a (0, 2) distribution, and where , and 2 are unknown

parameters.

Frequency

-0.4 -0.2 0.0 0.2 0.4 0.6 0.8

0 50 100 150 200

Residuals

6

(iii) Determine the fitted line of the regression model. [3]

(iv) (a) Identify which one of the following options gives the correct estimate of

the variance 2 of the model. [1]

(A1) 0.612

(A2) 1.098

(A3) 0.971

(A4) 0.139

(b) Identify which one of the following options gives the correct estimate of

the variance of the predicted mean response if x = 11. [2]

(A1) 0.161

(A2) 0.085

(A3) 0.287

(A4) 0.309

(c) Calculate a 95% confidence interval for the predicted mean response if

x = 11. [2]

(v) Comment on the width of a 95% confidence interval for the predicted mean

response if x = 3.5, as compared to the width of the interval in part (iv),

without calculating the new interval. [2]