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]