A-level · Data presentation

Correlation and regressionExam-style questions

Bivariate data and scatter diagrams; describing correlation by its direction (positive/negative) and strength; why correlation does not imply causation; and using a given regression line y = a + bx to make and interpret predictions — including why prediction inside the data range (interpolation) is safe but outside it (extrapolation) is not. Interpreting the PMCC is flagged as A-level.

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12 questions · A-level level · mark scheme and worked solution on each · no sign-up to try

DifficultyAll Level 2 Level 3 Level 4 Level 5

1
A running coach is studying the runners at her athletics club. For each of $10$ runners, she records the number of hours, $h$ , that the runner spends training in a typical week and the time, $m$ minutes, that the runner takes to complete a $5$ km race. The values of $h$ for these runners range from $3$ to $14$

Her results are summarised below.

$\sum h = 84 \qquad \sum m = 243.4 \qquad \sum h^2 = 840 \qquad \sum hm = 1945.8 \qquad S_{mm} = 79.904$
(a) Find the exact value of $S_{hh}$ and the exact value of $S_{hm}$

(3 marks)
(b) Calculate the product moment correlation coefficient between $h$ and $m$

(2 marks)
(c) Interpret your value of the product moment correlation coefficient in the context of these data.

(1 mark)
(d) Show that the equation of the regression line of $m$ on $h$ can be written as

$m = 30.5 - 0.735h$

where the values of the intercept and gradient are given to 3 significant figures.

(3 marks)
(e) Give an interpretation, in context, of the gradient of this regression line.

(1 mark)
(f) Using the equation in part (d),

(i) estimate the time taken to complete a $5$ km race by a runner who trains for $9$ hours each week,

(ii) explain why it would not be sensible to use the equation to estimate the time taken by a runner who trains for $25$ hours each week.

(2 marks)
Level 4 · 12 marksTry it
2
An energy consultant visits $11$ houses in a town. For each house, she records the floor area, $x\ \text{m}^2$ , and the amount of electricity used by the household in the previous year, $y$ MWh. Her results are shown in the table below.
House A B C D E F G H I J K
Floor area ( $x$ $\text{m}^2$ ) 62 75 84 90 98 110 118 130 140 156 172
Electricity used ( $y$ MWh) 2.3 2.6 3.1 3.0 3.5 3.6 4.0 9.6 4.4 4.9 5.3
The household in house H heats an outdoor swimming pool.

An outlier is defined as a value that is greater than $Q_3 + 1.5 \times (Q_3 - Q_1)$ or smaller than $Q_1 - 1.5 \times (Q_3 - Q_1)$
(a) Show that $9.6$ is an outlier for the amounts of electricity used.

(3 marks)

The consultant removes house H from the data. For the remaining $10$ houses,

$\sum x = 1105 \qquad \sum y = 36.7 \qquad S_{xx} = 11\,670.5 \qquad S_{xy} = 315.65$
(b) Show that, for these $10$ houses, the regression line of $y$ on $x$ has equation

$y = 0.681 + 0.0270x$

where the values of the intercept and gradient are given to 3 significant figures. You must show your working.

(3 marks)
(c) Give an interpretation of the gradient of this regression line.

(1 mark)

Two other houses in the town have floor areas of $125\ \text{m}^2$ and $250\ \text{m}^2$
(d) Use the equation in part (b) to estimate the amount of electricity used in a year by

(i) the house with a floor area of $125\ \text{m}^2$ ,

(ii) the house with a floor area of $250\ \text{m}^2$ .

(3 marks)
(e) State, giving a reason, which of your two estimates in part (d) is likely to be more reliable.

(2 marks)
Level 4 · 12 marksTry it
3
A courier firm records the distance, $x$ miles, and the time taken, $y$ minutes, for each of $15$ parcel deliveries.

The following summary statistics are available.

$\sum x = 81.4 \qquad \sum x^2 = 545.46 \qquad S_{yy} = 1110 \qquad S_{xy} = 322.7 \qquad n = 15$
(a) (i) Find the mean distance of these $15$ deliveries.

(ii) Show that the standard deviation of the distances is $2.63$ miles, correct to 3 significant figures.

(3 marks)
(b) Calculate the product moment correlation coefficient between $x$ and $y$

(3 marks)

The charge, £ $C$ , for delivering a parcel a distance of $x$ miles is worked out using the formula

$C = 4.50 + 0.90x$
(c) Using your answers to part (a), find the mean and the standard deviation of the charges for these $15$ deliveries. Give your answers to 3 significant figures.

(4 marks)
(d) State, giving a reason, the value of the product moment correlation coefficient between $C$ and $y$

(2 marks)
Level 4 · 12 marksTry it
4
A car dealer records the age, $x$ $x$ years, and the price, £ $y$ $y$ thousand, of each car in a random sample of $16$ $16$ used cars of the same model. Using these data, the dealer finds that
- the regression line of $y$ on $x$ is $\quad y = 18.4 - 1.6x$
- the regression line of $x$ on $y$ is $\quad x = 10.4 - 0.5y$
(a) (i) Find the coordinates of the point where these two lines meet.

(ii) Hence show that $\sum y = 140.8$ $\sum y = 140.8$

(4 marks)

The dealer also calculates that $\sum x^2 = 676$ $\sum x^{2} = 676$
(b) Find the value of $S_{xy}$ $S_{x y}$

(4 marks)
(c) Find the product moment correlation coefficient between $x$ $x$ and $y$ $y$

(4 marks)
Level 4 · 12 marksTry it
5
A meteorologist believes that windier days tend to be colder. She records the daily mean windspeed, $w$ knots, and the daily mean air temperature, $T\,^\circ$ C, on $10$ randomly chosen days at a weather station. The results are shown in the table.

$\begin{array}{|c|c|c|c|c|c|c|c|c|c|c|}\hline w & 4 & 6 & 7 & 9 & 10 & 12 & 14 & 15 & 18 & 20 \\ \hline T & 16.2 & 13.8 & 14.9 & 17.0 & 13.1 & 14.4 & 11.9 & 15.5 & 10.8 & 12.6 \\ \hline \end{array}$

For these data the product moment correlation coefficient is $r = -0.627$ and the equation of the regression line of $T$ on $w$ is $T = 16.7 - 0.234w$ .
(a) Test, at the $5\%$ level of significance, whether these data provide evidence to support the meteorologist's belief. State your hypotheses clearly. (4 marks)
(b) Give an interpretation of the gradient of the regression line in this context. (1 mark)
(c) Use the regression line to estimate the daily mean air temperature on a day when the daily mean windspeed is $12.5$ knots, giving your answer to $1$ decimal place. (2 marks)
(d) Explain why it would be unreliable to use this regression line to estimate (i) the daily mean air temperature on a day when the daily mean windspeed is $35$ knots, (ii) the daily mean windspeed on a day when the daily mean air temperature is $12\,^\circ$ C. (2 marks)
Level 4 · 9 marksTry it

House	A	B	C	D	E	F	G	H	I	J	K
Floor area ( $x$ $\text{m}^2$ )	62	75	84	90	98	110	118	130	140	156	172
Electricity used ( $y$ MWh)	2.3	2.6	3.1	3.0	3.5	3.6	4.0	9.6	4.4	4.9	5.3

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