The random variables \(X\), \(Y\) follow a bivariate normal distribution with product moment correlation coefficient \(\rho \). The following table gives a random sample from this distribution.

(a)     Determine the value of \(r\), the product moment correlation coefficient of this sample.

(b)     (i)     Write down hypotheses in terms of \(\rho \) which would enable you to test whether or not \(X\) and \(Y\) are independent.

(ii)     Determine the p-value of the above sample and state your conclusion at the 5% significance level. Justify your answer.

(c)     (i)     Determine the equation of the regression line of \(y\) on \(x\).

(ii)     State whether or not this equation can be used to obtain an accurate prediction of the value of \(y\) for a given value of \(x\). Give a reason for your answer.

(a)     \(r =  - 0.163\)     A2

[2 marks]

(b)     (i)     \({{\text{H}}_0}:\rho  = 0:{{\text{H}}_1}:\rho  \ne 0\)     A1

(ii)     \(t = r\sqrt {\frac{{n - 2}}{{1 - {r^2}}}}  =  - 0.468 \ldots \)     (A1)

\({\text{DF}} = 8\)     (A1)

\(p{\text{-value}} = 2 \times 0.326 \ldots  = 0.652\)   A1

since \(0.652 > 0.05\), we accept \({{\text{H}}_0}\)     R1

Note: Award (A1)(A1)A0 if the p-value is given as \(0.326\) without prior working.

Note: Follow through their p-value for the R1.

[5 marks]

(c)     (i)     \(y =  - 0.257x + 5.22\)     A1

Note: Accept answers which round to \(–0.26\) and \(5.2\).

(ii)     no, because \(X\) and \(Y\) have been shown to be independent (or equivalent)     A1

[2 marks]