GCSE Statistics · Higher · Chapter 10: Correlation and regression

Scatter diagrams, regression and correlation coefficients

About 85 minutesCalculator allowed75 marks

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Date

Do Now

The first two are what this chapter needs. The rest bring back earlier chapters so nothing is forgotten.

A1
Needed today
Find the gradient of the line through $(5, 2)$ and $(- 3, 0)$ .
[2 marks]
Answer

A2
Needed today
An overall mark is worked out from Paper 1, worth 35%, and Paper 2, worth 65%. Liam scored $93$ in Paper 1 and $57$ in Paper 2. Work out Liam's overall mark.
[2 marks]
Answer
A3
From chapter 2
There are 1800 fish in a lake. Sample A is 20 of them, and 9 are over 20 cm long. Sample B is 120 of them, and 32 are over 20 cm long. Use the more reliable sample to estimate how many of the 1800 are over 20 cm long, to the nearest whole number.
[2 marks]
Answer
A4
From chapter 2
Isla numbers 260 people from 1 to 260. The calculator's random number key gives $0.893$ . Isla multiplies it by 260 and rounds up to choose a person. Which person is chosen?
[2 marks]
Answer
A5
From chapter 2
Maya wants to find out how many students in a school own a pet. Maya plans a stratified sample by whether each student owns a pet. Is this a sensible variable to stratify by? Give a reason.
[2 marks]

Examples, then your turn

Your teacher works through each example with you. Then try the one beside it on your own. Spearman's rank correlation coefficient is $r_{s} = 1 - \frac{6 \sum d ^{2}}{n ( n ^{2} - 1 )}$ .

Example 1

For nine months, the scatter diagram shows how many adverts a shop placed in the local paper and how many customers it had. Describe the correlation, then say what it means for the shop.

Hint Name the type of correlation, then say what it means for adverts and customers.

B1
The scatter diagram shows the height above sea level and the temperature at eight weather stations at noon. Describe the correlation, and say what it shows about height and temperature.
[2 marks]

Example 2

The cooking time, $T$ minutes, for a chicken of mass $m$ kg is $T = 20 + 40 m$ . Interpret the $20$ and the $40$ in this context.

Hint The

40

is what is added for each extra kg. The

20

is what is there when

m = 0

B2
The monthly phone bill, £ $C$ , for $t$ minutes of calls is $C = 8 + 0.05 t$ . Interpret the $8$ and the $0.05$ in this context.
[2 marks]

Example 3

For laptops from 1 to 8 years old, the value, £ $y$ , is given by $y = 900 - 75 x$ , where $x$ is the age in years. Estimate the value of a 5-year-old laptop and of a 14-year-old laptop. Comment on the reliability of each.

Hint Substitute each age. Then check whether the age is inside the range of the data.

B3
For sunflowers from 2 to 10 weeks old, the height, $h$ cm, is given by $h = 12 + 9 w$ , where $w$ is the age in weeks. Estimate the height of a sunflower at 6 weeks and at 20 weeks. Which estimate is more reliable? Give a reason.
[3 marks]

Example 4

The table shows the population and the number of libraries in six towns. Work out Spearman's rank correlation coefficient, correct to 3 decimal places.

Hint Rank the populations, largest as 1. Rank the libraries the same way.

B4
Eight used cars, A to H. Ages in years: $1, 3, 5, 2, 7, 4, 6, 8$ . Prices in £: $9800, 6100, 5200, 8800, 3900, 7900, 3300, 2900$ . Find Spearman's rank correlation coefficient, to 2 decimal places.
[4 marks]
Answer

Example 5

Two judges each rank eight photographs. Should the PMCC or Spearman's rank correlation coefficient be used to measure how well they agree? Give a reason.

Hint Are the data measurements or ranks?

B5
The scatter diagram shows the temperature of a hot drink as it cools. Should the PMCC or Spearman's rank correlation coefficient be used to measure the correlation? Give a reason.
[2 marks]

Practice

Each question asks for something different. Show your working.

C1
The scatter diagram shows the length and mass of eight fish. Write down the length and mass of the fish that does not fit the pattern. A scientist checks this fish and finds the mass was written down as 600 g instead of 300 g. What should she do with this point?
[2 marks]

C2
The table shows six pairs of values. Work out the coordinates of the double mean point.
[2 marks]
Answer $(\dots\dots, \dots\dots)$
C3
A café records the temperature, $x$ °C, and the number of cold drinks sold, $y$ , on 20 days. The regression line is $y = a + 1.5 x$ . The mean temperature is $12$ °C and the mean number of cold drinks is $40$ . Work out the value of $a$ .
[2 marks]
Answer
C4
The table shows how two judges ranked six singers. Calculate $r_{s}$ for the two judges, to 3 decimal places.
[3 marks]
Answer
C5
For 12 students, Spearman's rank correlation coefficient between their height and their score in a spelling test is $0.04$ . Interpret this value in context.
[1 mark]
C6
A farmer records the rainfall each month and the mass of potatoes his fields produce. Which is the explanatory variable? Give a reason.
[2 marks]
C7
The number of fire engines sent to a fire and the cost of the damage show strong positive correlation. Lucy says, "Sending more fire engines causes more damage." Explain why Lucy is wrong.
[2 marks]
C8
The PMCC between the mass of some parcels and the cost of posting them is $0.86$ . Kat draws the scatter diagram again with the axes swapped, so the cost goes across and the mass goes up. Write down the PMCC for her new diagram.
[1 mark]
Answer

Exam-style questions

Answer every question in the space given. Show your working: most of the marks are for method.

D1
The table shows the engine size and the fuel economy of eight cars.
(a)
Work out the coordinates of the double mean point.
[2 marks]
Answer $(\dots\dots, \dots\dots)$
(b)
The regression line is $y = 60 - 10 x$ . Show that it passes through the double mean point.
[1 mark]
Answer
(c)
Interpret the gradient $- 10$ in context.
[1 mark]
(d)
Use the regression line to estimate the fuel economy of a car with a 2.6 litre engine. Comment on the reliability of your estimate.
[2 marks]
Total for D1: 6 marks

D2
For eight hospitals, Spearman's rank correlation coefficient between the number of nurses and the number of beds is $1$ . Raj says, "So the points on the scatter diagram must lie on a straight line."
Is Raj correct? Give a reason for your answer.
[2 marks]
D3
Mia's hypothesis is: "Gyms that charge more have fewer members." The table shows data for seven gyms.
(a)
Calculate Spearman's rank correlation coefficient for these gyms. Round to 2 decimal places.
[3 marks]
Answer
(b)
Does this value support Mia's hypothesis? Give a reason.
[2 marks]
Total for D3: 5 marks
D4
On days between $12$ °C and $28$ °C, a shop's daily sales of ice lollies, $y$ , and the temperature, $x$ °C, give the regression line $y = - 20 + 6 x$ .
(a)
Estimate how many ice lollies the shop sells on a $20$ °C day.
[1 mark]
Answer
(b)
Ben uses the line for a day when the temperature is $2$ °C and gets $- 8$ ice lollies. Give two reasons why this estimate is wrong.
[2 marks]
Total for D4: 3 marks
D5
Leo wants to find out whether taller students have longer arm spans. His plan is to measure the heights of 20 students in Year 10 and the arm spans of 20 students in Year 11, then work out the PMCC. Assess Leo's plan.
[3 marks]
D6
For adults with heights from $150$ cm to $190$ cm, the mass, $y$ kg, and the height, $x$ cm, give the regression line $y = - 40 + 0.6 x$ . Explain why the value $- 40$ has no meaning in this context.
[2 marks]
D7
For 12 towns, Spearman's rank correlation coefficient between population and the number of shops is $0.95$ , and the PMCC is $0.71$ . What do these two values tell you about the scatter diagram?
[2 marks]
D8
The scatter diagram shows the minutes of exercise a day and the resting heart rate of eight adults. The double mean point is $(30, 68)$ .
(a)
Describe the correlation. Interpret it in context.
[2 marks]
(b)
Draw a line of best fit that passes through $(30, 68)$ . Use it to estimate the resting heart rate of an adult who exercises for $35$ minutes a day.
[2 marks]
Answer
Total for D8: 4 marks

Extension

No route is given. Spend five minutes on a problem before you look at a hint. The hints are at the end of the sheet. Use one at a time.

E1
For 8 pairs of data, $\sum x = 96$ and $\sum y = 340$ . The regression line $y = a + b x$ has gradient $2.5$ . Work out $a$ , and estimate $y$ when $x = 14$ .
[3 marks]
Hint 1 · What to try
The regression line passes through the double mean point.
Hint 2 · The first line
$x ˉ = 96 \div 8 = 12$ and $y ˉ = 340 \div 8 = 42.5$ .
Hint 3 · The full method
$42.5 = a + 2.5 \times 12$ . Then substitute $x = 14$ .
Answer $a = \dots\dots$ $y = \dots\dots$

E2
Two judges rank 10 cakes. Judge B's ranks are Judge A's ranks with some pairs of neighbouring ranks swapped (for example ranks $4$ and $5$ swapped), and no rank is in more than one swap. Spearman's rank correlation coefficient is $0.964$ to 3 decimal places. How many swaps are there?
[4 marks]
Hint 1 · What to try
Swapping two neighbouring ranks gives two values of $d$ . What are they?
Hint 2 · The first line
Each swap gives $d = 1$ and $d = - 1$ , so it adds $2$ to $\sum d^{2}$ .
Hint 3 · The full method
With $k$ swaps, $r_{s} = 1 - \frac{6 \times 2 k}{10 \times 99}$ . Try $k = 1, 2, 3, \dots$
Answer
E3
Eight points have double mean point $(6, 20)$ and a PMCC of $0.6$ . A ninth point, $(6, 20)$ , is added. Write down the new PMCC, and explain why it does not change.
[3 marks]
Hint 1 · What to try
Work out the new double mean point.
Hint 2 · The first line
It is still $(6, 20)$ . How far is the new point from it?
Hint 3 · The full method
The PMCC is worked out from each point's distances from the mean point. The new point's distances are $0$ .

How did it go?

Colour one circle on each line.

	Not yet	Nearly	Yes
I can describe correlation from a scatter diagram in context, and I know it does not show cause.
I can use a regression line through the double mean point, interpret $a$ and $b$ , and judge a prediction.
I can work out Spearman's rank from raw data and interpret it in context.
I can interpret the PMCC and say when Spearman's rank is the better choice.

Answers and mark scheme

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