State suitable hypotheses to investigate whether or not a negative linear association exists between X and Y.

Determine the p-value.

State your conclusion at the 1 % significance level.

Show that Cov(U, V) = E(UV) − E(U)E(V).

Hence show that if U, V are independent random variables then the population product moment correlation coefficient, ρ, is zero.

* This question is from an exam for a previous syllabus, and may contain minor differences in marking or structure.

H₀: ρ = 0; H₁: ρ < 0       A1

[1 mark]

$t=-0.708\sqrt{\frac{11-2}{1-{\left(-0.708\right)}^2}}=\left(-3.0075\dots \right)$       (M1)

degrees of freedom = 9        (A1)

P(T < −3.0075...) = 0.00739       A1

Note: Accept any answer that rounds to 0.0074.

[3 marks]

reject H₀ or equivalent statement       R1

Note: Apply follow through on the candidate’s p-value.

[1 mark]

Cov(U, V) + E((U − E(U))(V − E(V)))

= E(UV − E(U)V − E(V)U + E(U)E(V))       M1

= E(UV) − E(E(U)V) − E(E(V)U) + E(E(U)E(V))       (A1)

= E(UV) − E(U)E(V) − E(V)E(U) + E(U)E(V)       A1

Cov(U, V) = E(UV) − E(U)E(V)       AG

[3 marks]

E(UV) = E(U)E(V) (independent random variables)       R1

⇒Cov(U, V) = E(U)E(V) − E(U)E(V) = 0      A1

hence, ρ = $\frac{\text{Cov}\left(U,V\right)}{\sqrt{\text{Var}\left(U\right)\text{Var}\left(V\right)}}=0$     A1AG

Note: Accept the statement that Cov(U,V) is the numerator of the formula for ρ.

Note: Only award the first A1 if the R1 is awarded.

[3 marks]

State suitable hypotheses $H_0$ and $H_1$ to test Peter’s claim, using a two-tailed test.

Carry out a suitable test at the 5 % significance level. With reference to the $p$-value, state your conclusion in the context of Peter’s claim.

Peter uses the regression line of $y$ on $x$ as $y=0.248x+83.0$ and calculates that a student with a Mathematics test score of 73 will have a running time of 101 seconds. Comment on the validity of his calculation.

$H_0\text{:}\rho =0$   $H_1\text{:}\rho \ne 0$       A1

Note: It must be $\rho$.

[1 mark]

$p=0.649$       A2

Note: Accept anything that rounds to 0.65

0.649 > 0.05        R1

hence, we accept $H_0$ and conclude that Peter’s claim is wrong         A1

Note: The A mark depends on the R mark and the answer must be given in context. Follow through the $p$-value in part (b).

[4 marks]

a statement along along the lines of ‘(we have accepted that) the two variables are independent’ or ‘the two variables are weakly correlated’       R1

a statement along the lines of ‘the use of the regression line is invalid’ or ‘it would give an inaccurate result’       R1

Note: Award the second R1 only if the first R1 is awarded.

Note: FT the conclusion in(a)(ii). If a candidate concludes that the claim is correct, mark as follows: (as we have accepted H₁) the 2 variables are dependent and 73 lies in the range of $x$ values R1, hence the use of the regression line is valid R1. 

[2 marks]