For these data find the product moment correlation coefficient, $r$.

(i)     State the distribution of the test statistic (including any parameters).

(ii)     Find the $p$-value for the test.

(iii)     State the conclusion, in the context of the question, with the word “correlation” in your answer. Justify your answer.

Using a suitable regression line, find an estimate for his score on Exam 2, giving your answer to the nearest integer.

(i)     State the distribution of your test statistic (including any parameters).

(ii)     Find the $p$-value.

(iii)     State the conclusion, justifying the answer.

$r = 0.804$    A2

Note: Accept any number that rounds to 0.80.

[2 marks]

(i)     $t$ distribution with 8 degrees of freedom     A1A1

(ii)     $p{\text{ - value}} = 0.00254$     A2

Notes: Accept any number that rounds to 0.0025.

Award A1 for 2-tail test giving an answer that rounds to 0.0051.

(iii)     $p{\text{ - value}} < 0.01$, so conclude that there is positive correlation     R1A1

Notes: Only award the A1 if the R1 is awarded.

Do not accept just “reject ${H_0}$” or “accept ${H_1}$”.

The words “positive correlation” must be seen.

[6 marks]

regression line of $y$ (Exam 2 mark) on $x$ (Exam 1 mark) is     (M1)

$y = 0.59407 \ldots x + 21.387 \ldots$    (A1)

$x = 11$ gives $y = 28$ (to nearest integer)     A1

[3 marks]

(i)     applying the $t$ test to the differences

$t$ distribution with 9 degrees of freedom     A1A1

(ii)     $p{\text{ - value}} = 0.239$     A2

Notes: Accept any number that rounds to 0.24.

Award A1 if subtraction done the wrong way round giving $p{\text{ - value}} = 0.109$.

(iii)     $p{\text{ - value}} > 0.05$, so accept ${H_0}$ or ${u_d} = 6$     R1A1

[6 marks]