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 teacher compares her students' results in a mock examination with their results in the final examination. For a random sample of $12$ of her students, she records the percentage mark in the mock examination, $x$ , and the percentage mark in the final examination, $y$ . The mock examination marks of these students range from $28$ to $85$

The data are summarised as follows.

$\sum x = 681 \qquad \sum y = 723 \qquad \sum x^2 = 42\,055 \qquad \sum y^2 = 45\,711 \qquad \sum xy = 43\,629$
(a) Find the values of $S_{xx}$ , $S_{yy}$ and $S_{xy}$

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

(2 marks)
(c) Give an interpretation of your product moment correlation coefficient.

(1 mark)
(d) Show that the equation of the regression line of $y$ on $x$ is

$y = 17.0 + 0.762x$

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 the regression line.

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

(i) estimate the final examination mark of a student who scored $60$ in the mock examination,

(ii) explain why the equation should not be used to estimate the final examination mark of a student who scored $15$ in the mock examination.

(2 marks)
(g) Using the equation in part (d), find the range of values of $x$ for which a student is predicted to score a higher mark in the final examination than in the mock examination.

(2 marks)
Level 5 · 14 marksTry it
2
Nadia is a letting agent in a city. She wants to model the relationship between the distance, $d$ km, of a one-bedroom flat from the city centre and the monthly rent, £ $p$ , of the flat.
(a) Give one advantage of using a statistical model to describe a real-world situation such as this.

(1 mark)

Nadia takes a random sample of $12$ one-bedroom flats that she lets. She codes the monthly rents using

$w = \frac{p - 600}{10}$

Her data are summarised below.

$\sum d = 54 \qquad \sum w = 589 \qquad \sum w^2 = 33\,549 \qquad S_{dd} = 70.12 \qquad S_{dw} = -552.4$
(b) Show that $S_{ww} = 4638.92$ to 2 decimal places.

(1 mark)
(c) Use the coding to find the value of $S_{dp}$ and the value of $S_{pp}$

(3 marks)
(d) Find the product moment correlation coefficient between $d$ and $p$

(2 marks)
(e) Interpret this value in the context of the question.

(1 mark)
(f) Show that the equation of the regression line of $w$ on $d$ is

$w = 84.5 - 7.88d$

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

(3 marks)
(g) Hence find the equation of the regression line of $p$ on $d$ , giving your answer in the form $p = a + bd$

(1 mark)
(h) Using your equation from part (g),

(i) estimate the monthly rent of a one-bedroom flat that is $4$ km from the city centre,

(ii) state the change in monthly rent that the model predicts for each extra kilometre from the city centre.

(2 marks)
Level 5 · 14 marksTry it
3
An engineer is investigating how the stopping distance of a car depends on its speed. She tests a car on a dry road at $12$ different speeds. For each test she records the speed of the car, $x\ \text{km h}^{-1}$ , and its stopping distance, $y$ metres, where $20 \leqslant x \leqslant 130$

The equation of the regression line of $y$ on $x$ for her data is

$y = -22.2 + 1.01x$
(a) Give an interpretation of the gradient of this regression line.

(1 mark)
(b) Use the regression line to estimate the stopping distance of the car when it is travelling at $150\ \text{km h}^{-1}$

(2 marks)
(c) State, giving a reason, whether your estimate in part (b) is likely to be reliable.

(2 marks)

The engineer also calculates the following summary statistics.

$\sum y = 551.7 \qquad \sum y^2 = 40\,583.05 \qquad \sum x^2 = 69\,150 \qquad n = 12$
(d) Using these summary statistics and the equation of the regression line, show that the product moment correlation coefficient between $x$ and $y$ is $0.985$ to 3 decimal places.

(7 marks)

The engineer's results are shown on the scatter diagram below.(e) Using the scatter diagram and the product moment correlation coefficient, discuss whether a linear regression model is suitable for describing how stopping distance depends on speed.

(2 marks)
Level 5 · 14 marksTry it