Correlation and Causation (Higher) - Worksheets, Questions and Revision

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HIGHER

H02 Correlation and Causation (Higher)

EDEXCEL 1ST0 · Calculator allowed · about 90 minutes

Total Marks

Name: _______________________________ Date: ____ / ____ / ______

Answer ALL questions. Show all your working.

A cafe owner records the number of hours of sunshine each afternoon and the number of hot chocolates sold that afternoon, over 8 different afternoons. The scatter graph shows the results.

(a)Write down the type of correlation shown by the scatter graph.(1)

(b)Describe what the scatter graph shows about the relationship between hours of sunshine and the number of hot chocolates sold.(2)

(c)Suggest one reason why this information could be useful to the cafe owner.(1)

(Total for Question 1 is 4 marks)

For each statement about correlation and lines of best fit, write down whether it is True or False.

(i)If two variables have a strong correlation, this proves that one variable causes changes in the other.(1)

(ii)A confounding variable is a factor, other than the two variables being studied, that could explain why they appear to be related.(1)

(iii)Using a line of best fit to estimate a value for x that lies outside the range of the original data is called interpolation.(1)

(iv)Estimates made by extrapolating a line of best fit are generally less reliable than estimates made by interpolating.(1)

(Total for Question 2 is 4 marks)

For each pair of variables, write down whether you would expect positive correlation, negative correlation, or no correlation.

(a)The number of pages in a paperback book and the price of the book.(1)

(b)The age of a second-hand car and its resale value.(1)

(c)A student's shoe size and their mark in a geography test.(1)

(Total for Question 3 is 3 marks)

A shopkeeper records the midday temperature and the number of cold drinks sold on 7 different days. The scatter graph shows the results, together with a line of best fit.

(a)Use the line of best fit to estimate the number of cold drinks sold when the temperature is 18 degrees C.(2)

(b)State whether your estimate in part (a) was found by interpolation or extrapolation, and explain whether you would expect it to be reliable.(2)

(c)Give one reason why this estimate might still not be exactly correct.(1)

(Total for Question 4 is 5 marks)

A researcher records the distance (in miles) and the cost (in £) of several taxi journeys. She plots the results on a scatter graph and draws a line of best fit. Two points that lie exactly on this line of best fit are (2, 7) and (10, 23), where x is the distance in miles and y is the cost in £.

(a)Work out the gradient of the line of best fit.(2)

(b)Find the equation of the line of best fit in the form y = mx + c.(2)

(c)Use the equation to estimate the cost of a 6-mile taxi journey.(1)

(d)Explain why using this equation to estimate the cost of a 50-mile taxi journey would be unreliable.(1)

(Total for Question 5 is 6 marks)

The formula for the product moment correlation coefficient (PMCC), r, is:

r = Sxy / √Sxx * Syy

where Sxy = (sum of xy) - (sum of x)(sum of y)/n, Sxx = (sum of x²) - (sum of x)²/n and Syy = (sum of y²) - (sum of y)²/n.

For a set of paired data, Sxy = 84, Sxx = 120 and Syy = 98.

Work out the value of r, giving your answer correct to 3 significant figures.

(Total for Question 6 is 3 marks)

The table shows the product moment correlation coefficient, r, calculated between two variables in four different studies.

Study	r
A	-0.92
B	0.15
C	0.68
D	-0.40

(a)Write down which study shows the strongest association between its two variables, regardless of sign.(1)

(b)Write down which study shows the weakest association between its two variables.(1)

(c)Write down which studies show a negative association.(1)

(d)Describe the direction and strength of the association shown in Study C.(1)

(Total for Question 7 is 4 marks)

A teacher records the number of hours 5 pupils spent revising for a test, x, and their test score, y. The table and scatter graph show the results.

Pupil	Hours revising, x	Test score, y
1	2	48
2	4	62
3	5	52
4	7	74
5	9	80

You may use: Sxy = (sum of xy) - (sum of x)(sum of y)/n, Sxx = (sum of x²) - (sum of x)²/n, Syy = (sum of y²) - (sum of y)²/n, and r = Sxy / √Sxx * Syy.

(a)Show that Sxy = 135.6, Sxx = 29.2 and Syy = 756.8.(3)

(b)Hence calculate the value of r, giving your answer correct to 3 significant figures.(2)

(c)Interpret the value of r in the context of hours spent revising and test score.(2)

(Total for Question 8 is 7 marks)

In a seaside town, a student collects data over 12 months and finds a strong positive correlation (r = 0.89) between the average number of sunglasses sold per week and the average number of visits to the town's outdoor swimming pool per week.

(a)State the type of correlation described.(1)

(b)Give the name for a variable, such as weather, that could explain a correlation like this without either variable causing the other.(1)

(c)Explain why this correlation does not prove that buying sunglasses causes people to visit the swimming pool more often.(2)

(d)Suggest one thing the student could do to check whether weather is really responsible for the correlation.(1)

(Total for Question 9 is 5 marks)

For 7 employees at a company, x is the number of years they have worked there, and y is the total number of minutes they arrived late to work last month. The scatter graph shows the results.

(a)Describe the pattern shown by the scatter graph.(1)

(b)The product moment correlation coefficient for this data is r = -0.618. State whether this value is consistent with the pattern shown.(1)

(c)Explain your answer to part (b).(2)

(Total for Question 10 is 4 marks)

A study of many different countries finds a positive correlation between the average number of mobile phones per 100 people and the average life expectancy (in years) in each country.

(a)State the type of correlation described.(1)

(b)Suggest two different factors, other than owning a mobile phone, that could help explain why countries with more mobile phones also tend to have higher life expectancy.(2)

(c)Explain why these factors make it invalid to conclude that owning a mobile phone directly increases how long people live.(2)

(d)The researcher wants stronger evidence about whether mobile phone ownership has its own effect. Suggest one way she could adjust her analysis to check this.(1)

(Total for Question 11 is 6 marks)

Over a 10-year period in a small town, records show a strong positive correlation (r = 0.95) between the number of new coffee shops opened each year and the number of new yoga studios opened each year.

(a)State the strength and direction of the correlation described.(1)

(b)Give a reason why it is very unlikely that opening more coffee shops directly causes more yoga studios to open (or vice versa).(1)

(c)Suggest what is more likely to be driving both numbers upward over the 10 years.(1)

(d)Explain why this example is best described as a spurious (coincidental) correlation, rather than showing a genuine causal link.(2)

(Total for Question 12 is 5 marks)

Hospital records show a positive correlation between the number of nurses assigned to a hospital ward and the number of patient falls recorded on that ward. A hospital manager claims: 'Assigning more nurses to a ward causes more patient falls, so wards should be given fewer nurses.'

(a)State the type of correlation described.(1)

(b)Explain why the manager's conclusion may have the direction of cause and effect the wrong way round.(2)

(c)Suggest what is more likely to be the actual explanation for this data.(2)

(Total for Question 13 is 5 marks)

A researcher records, for 6 teenagers, the number of hours of TV watched per week, x, and their score in a fitness test out of 50, y.

Teenager	Hours of TV, x	Fitness score, y
1	5	44
2	8	40
3	10	30
4	14	32
5	18	24
6	21	16

You may use: Sxy = (sum of xy) - (sum of x)(sum of y)/n, Sxx = (sum of x²) - (sum of x)²/n, Syy = (sum of y²) - (sum of y)²/n, and r = Sxy / √Sxx * Syy.

(a)Calculate Sxy, Sxx and Syy for this data.(3)

(b)Hence calculate the value of r, giving your answer correct to 3 significant figures.(2)

(c)Interpret the value of r in the context of hours of TV watched and fitness test score.(2)

(Total for Question 14 is 7 marks)

A researcher wants to investigate the relationship between the amount of time teenagers spend on screens each evening and how long they sleep that night. She plans her investigation using the four stages of the statistical enquiry cycle: plan, collect, process and interpret.

(a)(Plan) Explain why the researcher should collect screen time and sleep duration as paired data from the same teenagers, rather than collecting each separately from different groups.(1)

(b)(Collect) The researcher only collects data from volunteers at one school. Give one reason why this might make her results unreliable for teenagers in general.(1)

(c)(Process) The scatter graph of the data shows a roughly linear pattern. State which measure, r or a visual description alone, gives a more precise summary of the strength of a linear relationship, and give one reason why.(2)

(d)(Interpret) The researcher calculates r = -0.7 for her data. Interpret this value, and explain why it would not be correct to conclude that screen time directly causes shorter sleep.(2)

(Total for Question 15 is 6 marks)

A national newspaper reports: 'New study finds r = 0.97 correlation between a country's average chocolate consumption per person and its number of universities that produce prize-winning scientific research. Eating chocolate boosts brainpower, say researchers.'

Evaluate this claim.

(Total for Question 16 is 6 marks)

Mark scheme · H02 Correlation and Causation

Question 1

(a) B1 Negative cao
(a) Answer: Negative correlation
(b) B1 identifies that as hours of sunshine increases, hot chocolate sales decrease, oe
(b) B1 correct context, e.g. sunnier afternoons are linked with fewer hot chocolates being sold
(b) Answer: As the number of hours of sunshine increases, the number of hot chocolates sold tends to decrease.
(c) B1 a sensible practical use, e.g. to help decide how much hot chocolate stock to prepare based on the weather forecast, oe
(c) Answer: It could help her decide how much hot chocolate to prepare based on the weather forecast (less on sunny days).

Question 2

(i) B1 False cao
(i) Answer: False
(ii) B1 True cao
(ii) Answer: True
(iii) B1 False cao
(iii) Answer: False
(iv) B1 True cao
(iv) Answer: True

Question 3

(a) B1 Positive cao
(a) Answer: Positive correlation
(b) B1 Negative cao
(b) Answer: Negative correlation
(c) B1 No correlation cao
(c) Answer: No correlation

Question 4

(a) M1 reads from the line of best fit at x = 18
(a) A1 awrt 34 (accept 32 to 36)
(a) Answer: 34 (accept 32-36)
(b) B1 Interpolation cao
(b) B1 reliable/fairly reliable because x = 18 lies within the range of the recorded temperatures (14 to 26), oe
(b) Answer: Interpolation; it should be fairly reliable because 18 degrees C lies within the range of the temperatures recorded (14 to 26 degrees C).
(c) B1 the data points do not all lie exactly on the line, so any single estimate may differ from the true value, oe
(c) Answer: The actual data points do not lie exactly on the line of best fit, so the true value could be a little higher or lower.

Question 5

(a) M1 (23-7)/(10-2), oe
(a) A1 2 cao
(a) Answer: 2
(b) M1 substitutes gradient and one point to find c, e.g. 7 = 2(2) + c
(b) A1 y = 2x + 3 cao
(b) Answer: y = 2x + 3
(c) B1 £15 cao, ft from (b)
(c) Answer: £15
(d) B1 50 miles is well outside the range of the original data (2 to 10 miles), so this is extrapolation and the linear pattern may not continue, e.g. taxis may charge at a different rate for very long journeys, oe
(d) Answer: 50 miles is far outside the 2-10 mile range of the original data, so this is extrapolation - the fare may not keep increasing at the same steady rate for much longer journeys.

Question 6

M1 substitutes correctly into r = Sxy / √Sxx * Syy
A1 r = 0.774596... (or equivalent unrounded value)
A1 0.775 (3 sf), ft
Answer: 0.775

Question 7

(a) B1 A cao
(a) Answer: A
(b) B1 B cao
(b) Answer: B
(c) B1 A and D cao
(c) Answer: A and D
(d) B1 positive, moderate to fairly strong, oe
(d) Answer: A moderate to fairly strong positive association.

Question 8

(a) B1 Sxy = 1842 - (27)(316)/5 = 135.6
(a) B1 Sxx = 175 - 27²/5 = 29.2
(a) B1 Syy = 20728 - 316²/5 = 756.8
(a) Answer: Sxy = 135.6, Sxx = 29.2, Syy = 756.8
(b) M1 r = 135.6 / √29.2 * 756.8, oe
(b) A1 0.912 (3 sf), ft from (a)
(b) Answer: 0.912
(c) B1 identifies a strong positive correlation/association
(c) B1 correct context, e.g. pupils who revised for longer tended to get higher test scores, oe
(c) Answer: There is a strong positive correlation - pupils who spent longer revising tended to achieve higher test scores.

Question 9

(a) B1 Positive (strong) cao
(a) Answer: Strong positive correlation
(b) B1 confounding variable / third variable, oe
(b) Answer: A confounding (or third) variable
(c) B1 identifies that a third factor (warm/sunny weather) could increase both sunglasses sales and pool visits independently
(c) B1 correct conclusion, e.g. the correlation could be explained by weather rather than one variable causing the other
(c) Answer: Warm, sunny weather could independently increase both sunglasses sales and pool visits, so the correlation may simply reflect the weather rather than one thing causing the other.
(d) B1 e.g. also record temperature/hours of sunshine each week and compare weeks with similar weather to see whether the link remains, oe
(d) Answer: Record the weather (e.g. temperature) each week too, and compare weeks with similar weather to see if the link between sunglasses sales and pool visits weakens.

Question 10

(a) B1 a generally downward/falling pattern with a fair amount of scatter, oe
(a) Answer: The points generally fall from left to right, but with a fair amount of scatter.
(b) B1 Yes cao
(b) Answer: Yes, it is consistent.
(c) B1 the negative sign matches the falling pattern of the points
(c) B1 the size of r (moderate, not close to -1) matches the amount of scatter around the trend rather than a perfect straight line
(c) Answer: The negative sign matches the overall falling trend, and because -0.618 is only moderate in size (not close to -1), this matches the noticeable scatter around that trend rather than a perfect line.

Question 11

(a) B1 Positive cao
(a) Answer: Positive correlation
(b) B1 national wealth/GDP per person, oe
(b) B1 a second distinct sensible factor, e.g. access to good healthcare, clean water and sanitation, or education levels, oe
(b) Answer: For example: national wealth (GDP per person), and access to good healthcare.
(c) B1 identifies that wealth (or another confounding factor) could cause both higher phone ownership and higher life expectancy
(c) B1 correct conclusion, e.g. the correlation may be entirely explained by wealth/healthcare rather than phones themselves having any effect on lifespan
(c) Answer: Wealth (or healthcare access) could independently cause both higher mobile phone ownership and higher life expectancy, so the correlation may have nothing to do with phones themselves affecting lifespan.
(d) B1 e.g. compare only countries with similar GDP/wealth and see whether the correlation between phone ownership and life expectancy still holds among just those countries, oe
(d) Answer: Compare only countries with similar levels of wealth, and see whether mobile phone ownership still correlates with life expectancy within that group.

Question 12

(a) B1 Strong positive cao
(a) Answer: Strong positive
(b) B1 there is no sensible/plausible mechanism linking the two business types directly, oe
(b) Answer: There is no realistic way that opening a coffee shop would directly cause someone to open a yoga studio, or vice versa.
(c) B1 a general trend affecting the town over the period, e.g. rising population, incomes, or overall high street development, oe
(c) Answer: A general trend in the town, such as a growing population or rising incomes, is more likely to be driving both types of new business.
(d) B1 identifies that both variables are simply increasing together over time due to an unrelated common trend, not because of any real link between coffee shops and yoga studios
(d) B1 correct conclusion, e.g. two unrelated quantities can appear strongly correlated purely because both happen to be rising over the same period
(d) Answer: Both quantities are simply rising together over the same 10 years because of the town's general growth, not because one type of business affects the other - two unrelated things can show a strong correlation purely by coincidence when both are increasing over time.

Question 13

(a) B1 Positive cao
(a) Answer: Positive correlation
(b) B1 identifies that it may be the level of patient risk/need that determines nurse numbers, not the other way round
(b) B1 correct explanation, e.g. wards with higher-risk or more frail patients are given more nurses, and these same patients are more likely to fall regardless of staffing
(b) Answer: It is more likely that wards with higher-risk, more frail patients are given more nurses in the first place, and these same patients are more likely to fall anyway - so patient risk may be causing both higher nurse numbers and more falls, not nurse numbers causing falls.
(c) B1 identifies patient frailty/risk level (or similar) as a factor affecting both nurse numbers and fall risk
(c) B1 expresses this as a case of reverse causation (fall risk influences staffing, rather than staffing influencing fall risk), oe
(c) Answer: Wards are likely staffed with more nurses because their patients are already more frail or at higher risk of falling - so the higher risk of falling is causing the higher nurse numbers (reverse causation), rather than nurse numbers causing the falls.

Question 14

(a) B1 Sxy = -300
(a) B1 Sxx = 187.3 (awrt), or 562/3
(a) B1 Syy = 526
(a) Answer: Sxy = -300, Sxx = 187.3 (3 sf), Syy = 526
(b) M1 r = -300 / √187.33... * 526, oe, ft from (a)
(b) A1 -0.956 (3 sf)
(b) Answer: -0.956
(c) B1 identifies a strong negative correlation/association
(c) B1 correct context, e.g. teenagers who watched more TV tended to get lower fitness scores, oe
(c) Answer: There is a strong negative correlation - teenagers who watched more hours of TV per week tended to get lower fitness test scores.

Question 15

(a) B1 paired data from the same individuals is needed to see whether screen time and sleep duration are linked for each person, so a correlation can be calculated, oe
(a) Answer: Correlation can only be measured between two variables recorded for the same individuals, so both measurements must come from the same teenagers.
(b) B1 one school is not representative of all teenagers (e.g. differences in area/school rules), and volunteers may not be typical, oe
(b) Answer: Teenagers at one school (and volunteers in particular) may not be representative of teenagers generally, e.g. because of differences in school rules or the type of pupils who volunteer.
(c) B1 r (the product moment correlation coefficient) cao
(c) B1 gives a reason, e.g. r gives an exact numerical value that can be compared precisely, whereas a visual description is subjective, oe
(c) Answer: r (the product moment correlation coefficient); it gives a precise numerical value, whereas describing a graph by eye is more subjective.
(d) B1 correctly interprets r = -0.7 as a fairly strong negative correlation, e.g. teenagers with more screen time tend to sleep less
(d) B1 identifies that this does not prove causation, e.g. a confounding variable such as a late bedtime routine or high stress/workload could affect both screen time and sleep duration
(d) Answer: r = -0.7 shows a fairly strong negative correlation - more screen time is linked with less sleep - but this does not prove screens cause poor sleep, since a confounding factor (e.g. a generally late bedtime routine, or high stress/workload) could independently affect both.

Question 16

B1 correctly reads r = 0.97 as a very strong positive correlation between the two variables
B1 states that even a very strong correlation does not by itself establish causation
B1 identifies a plausible confounding variable, e.g. national wealth/investment in research, which could independently raise both chocolate consumption and university research funding
B1 notes that the data is collected at the level of whole countries, so it says nothing about whether an individual person eating more chocolate improves their own brainpower
B1 suggests what further evidence or research design would be needed to test a causal claim properly, e.g. controlling for national wealth, or a controlled experiment on individuals
B1 reaches a balanced overall conclusion that the newspaper's causal claim is not supported by the correlation alone
Answer: The claim is not supported: r = 0.97 shows a very strong link between the two country-level variables, but wealthier countries can likely afford both more chocolate and better-funded universities, so national wealth is a plausible confounding variable; the correlation also says nothing about individual people, so it does not show that eating chocolate boosts anyone's own brainpower.

Question 16

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