Scatter diagrams, correlation and regressionAQA A-Level Maths: Subtopic test

1. 1
   A teacher records, for 25 students, the hours per week $h$ spent revising and the test score $s$ out of 100. The scatter diagram shows positive correlation and the regression line of $s$ on $h$ is $s=34+4.5h$. The values of $h$ range from 1 to 12.

   (a)

   Which statement about the value 4.5 in the regression line is correct?

   [1 mark]

   • AA student who does not revise scores 4.5
   • BEach extra hour of revision causes the score to rise by exactly 4.5
   • C4.5% of students revise for more than 12 hours
   • DEach extra hour of revision is associated with an increase of 4.5 in the score, on average

   (b)

   Use the line to estimate the score of a student who revises for 8 hours.

   [1 mark]

   • A70
   • B36
   • C38.5
   • D272

   (c)

   A student claims that revising for 30 hours a week would give a score of $34+4.5\times30$. Explain why this prediction is unreliable.

   [2 marks]

   Total for question 1: 4 marks

2. 2
   A scatter diagram shows the monthly sales of ice cream and the monthly number of beach rescues for 24 months in a seaside town. There is strong positive correlation between the two variables.

   (a)

   Which statement describes strong positive correlation?

   [1 mark]

   • AThe points lie close to a line sloping downwards
   • BThe points are scattered with no pattern
   • CThe points lie close to a line sloping upwards, so as sales rise rescues tend to rise
   • DEvery increase in sales causes an increase in rescues

   (b)

   Which is the most likely explanation of the correlation?

   [1 mark]

   • AEating ice cream causes people to need rescuing
   • BHot weather leads to both more ice-cream sales and more people swimming
   • CRescues cause an increase in ice-cream sales
   • DThe correlation must be a coincidence

   (c)

   A councillor proposes banning ice-cream sales to reduce the number of rescues. Comment on this proposal.

   [2 marks]

   Total for question 2: 4 marks

3. 3
   A scatter diagram shows the mass (kg) against the height (cm) of 60 adults. There is strong positive correlation overall, but the points form two distinct clusters: a lower cluster of 28 points and an upper cluster of 32 points. Further inspection shows that the lower cluster is the women and the upper cluster is the men.

   (a)

   Explain why a single regression line fitted to all 60 adults could be misleading.

   [3 marks]

   (b)

   The regression line for the women (heights 150 to 175 cm) is $m=0.75h-66.5$. The regression line for the men (heights 165 to 195 cm) is $m=1.1h-112$, where $m$ is the mass in kg and $h$ is the height in cm.
   (i) Interpret the gradient of the men's line.
   (ii) Compare the gradients of the two lines.
   (iii) Explain why the intercept of the men's line has no practical meaning.
   (iv) Explain why the women's line should not be used to estimate the mass of a woman of height 185 cm.

   [4 marks]

   Total for question 3: 7 marks

4. 4
   On 40 summer days a city council records the mean daily temperature $T$ ($^\circ$C), which ranges from 15 to 32, and the number $N$ of electric scooters hired. The scatter diagram shows strong positive correlation and the regression line of $N$ on $T$ is $N=12T-120$. The 10 weekend days form a separate cluster above the other 30 points.

   (a)

   (i) Interpret the gradient of the regression line.
   (ii) Explain why the intercept has no practical meaning.
   (iii) Use the line to estimate the number of hires on a day with mean temperature $25^\circ$C.
   (iv) Explain why the line should not be used for a day with mean temperature $10^\circ$C.

   [6 marks]

   (b)

   (i) Explain what the separate cluster of weekend days suggests, and how it affects estimates made with the single regression line.
   (ii) A councillor says: 'The strong correlation proves that warm weather causes people to hire scooters.' Evaluate this claim.
   (iii) Suggest two ways in which the council could improve its investigation.

   [6 marks]

   Total for question 4: 12 marks

End of questions