Question
Peter, the Principal of a college, believes that there is an association between the score in a Mathematics test, $X$, and the time taken to run $500 \mathrm{~m}, Y$ seconds, of his students. The following paired data are collected.
It can be assumed that $(X, Y)$ follow a bivariate normal distribution with product moment correlation coefficient $\rho$.
a.i. State suitable hypotheses $H_0$ and $H_1$ to test Peter’s claim, using a two-tailed test.
$[1]$
a.ii.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.
$[4]$
b. Peter uses the regression line of $y$ on $x$ as $y=0.248 x+83.0$ 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.
▶️Answer/Explanation
Markscheme
a.i. $H_0: \rho=0 \quad H_1: \rho \neq 0 \quad \boldsymbol{A 1}$
Note: It must be $\rho$.
[1 mark]
a.ii $p=0.649$
A2
Note: Accept anything that rounds to 0.65
$$
0.649>0.05 \quad R 1
$$
hence, we accept $H_0$ and conclude that Peter’s claim is wrong
$A 1$
Note: The $\boldsymbol{A}$ mark depends on the $\boldsymbol{R}$ mark and the answer must be given in context. Follow through the $p$-value in part (b).
[4 marks]
b. 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 $R 1$ only if the first $R 1$ is awarded.
Note: $\mathrm{FT}$ the conclusion in(a)(ii). If a candidate concludes that the claim is correct, mark as follows: (as we have accepted $\mathrm{H}_1$ ) the 2 variables are dependent and 73 lies in the range of $x$ values $\boldsymbol{R 1}$, hence the use of the regression line is valid $\boldsymbol{R} 1$.
[2 marks]
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Question
The random variables X , Y follow a bivariate normal distribution with product moment correlation coefficient ρ.
A random sample of 11 observations on X, Y was obtained and the value of the sample product moment correlation coefficient, r, was calculated to be −0.708.
The covariance of the random variables U, V is defined by
Cov(U, V) = E((U − E(U))(V − E(V))).
State suitable hypotheses to investigate whether or not a negative linear association exists between X and Y.[1]
Determine the p-value.[3]
State your conclusion at the 1 % significance level.[1]
Show that Cov(U, V) = E(UV) − E(U)E(V).[3]
Hence show that if U, V are independent random variables then the population product moment correlation coefficient, ρ, is zero.[3]
▶️Answer/Explanation
Markscheme
H0 : ρ = 0; H1 : ρ < 0 A1
[1 mark]
\(t = – 0.708\sqrt {\frac{{11 – 2}}{{1 – {{\left( { – 0.708} \right)}^2}}}} \,\, = \,\,\left( { – 3.0075 \ldots } \right)\) (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]
Examiners report
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