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Date	May 2018	Marks available	1	Reference code	18M.3.AHL.TZ0.Hsp_4
Level	Additional Higher Level	Paper	Paper 3	Time zone	Time zone 0
Command term	State	Question number	Hsp_4	Adapted from	N/A

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]

b.i.

State your conclusion at the 1 % significance level.

[1]

b.ii.

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

[3]

c.i.

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

[3]

c.ii.

Markscheme

* 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]

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

degrees of freedom = 9 (A1)

P(T < −3.0075...) = 0.00739 A1

Note: Accept any answer that rounds to 0.0074.

[3 marks]

b.i.

reject H₀ or equivalent statement R1

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

[1 mark]

b.ii.

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]

c.i.

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]

c.ii.

Examiners report

[N/A]

b.i.

[N/A]

b.ii.

[N/A]

c.i.

[N/A]

c.ii.

Syllabus sections

Topic 4—Statistics and probability » SL 4.4—Pearsons, scatter diagrams, eqn of y on x

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Find the exact value of m and of c for these data.
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Find the equation of the regression line of $y$ on $x$ .
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Show that the point M ( ${\bar x}$ , ${\bar y}$ ) lies on the regression line y on x.
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Draw the regression line on the scatter diagram.
21N.2.SL.TZ0.1a:
Find Pearson’s product-moment correlation coefficient, $r$ , for these data.
21N.2.AHL.TZ0.1a:
Find Pearson’s product-moment correlation coefficient, $r$ , for these data.
21N.2.AHL.TZ0.1e:
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State how, if at all, the value of $r$ would be affected.
17M.2.SL.TZ1.T_1a:
Find the range of the average body weights for these seven species of mammal.
17M.2.SL.TZ1.T_1c:
Write down the equation of the regression line $y$ on $x$ , in the form $y = mx + c$ .
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Use your regression line to estimate the average weight of the brain of grey wolves.
17M.2.SL.TZ1.T_1e:
Find the percentage error in your estimate in part (d).
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State whether it is valid to use the regression line to estimate the average weight of the brain of mice. Give a reason for your answer.
18N.2.SL.TZ0.T_1c.ii:
Justify whether it is valid to use the regression line y on x to estimate Jerome’s examination score.
18N.2.SL.TZ0.T_1d:
In his final IB examination Jerome scored 65.

Calculate the percentage error in Jerome’s estimated examination score.
16N.2.SL.TZ0.T_1a:
On graph paper, draw a scatter diagram for these data. Use a scale of 2 cm to represent 5 hours on the $x$ -axis and 2 cm to represent 10 points on the $y$ -axis.
16N.2.SL.TZ0.T_1b:
(i) ${\bar x}$ , the mean number of hours spent on social media;

(ii) ${\bar y}$ , the mean number of IB Diploma points.
16N.2.SL.TZ0.T_1f:
Draw the regression line, from part (e), on your scatter diagram.
16N.2.SL.TZ0.T_1g:
Use the given equation of the regression line to estimate the number of IB Diploma points that this girl obtained.
17M.2.SL.TZ2.T_3c:
Label the point ${\text{M}}(\bar x,{\text{ }}\bar C)$ on the scatter diagram.
17M.2.SL.TZ2.T_3e:
State a reason why the regression line $C$ on $x$ is appropriate to model the relationship between these variables.
17M.2.SL.TZ2.T_3g:
Draw the regression line $C$ on $x$ on the scatter diagram.
17M.2.SL.TZ2.T_3h:
Use the equation of the regression line to estimate the least number of folders that the factory needs to sell in a month to exceed its production cost for that month.

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Topic 4—Statistics and probability

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