A scatter graph shows a strong negative correlation between two variables. Which statement correctly describes this?
A) As one variable increases, the other variable also increases.
B) As one variable increases, the other variable decreases.
C) There is no relationship between the two variables.
D) Both variables always stay the same value.
(Total for Question 1 is 1 mark)
2
Priya records the outdoor temperature and the number of hot chocolates sold at her cafe each day. As the temperature increases, the number of hot chocolates sold decreases. State the type of correlation between temperature and the number of hot chocolates sold.
(Total for Question 2 is 1 mark)
3
For each situation below, state whether you would expect positive correlation, negative correlation or no correlation between the two quantities.
(a)The number of pages in a book and the weight of the book.(1)
(b)The age of a car and its value in pounds sterling.(1)
(c)A person's shoe size and their favourite colour.(1)
(Total for Question 3 is 3 marks)
4
A small company records the number of employees and its monthly revenue (in £000s). Number of employees (x): 2, 4, 5, 7, 9, 10, 12, 14, 15, 18. Monthly revenue in £000s (y): 18, 25, 30, 42, 50, 55, 65, 72, 80, 90.
(a)Plot all 10 points on the scatter graph grid.(2)
(b)Describe the correlation shown by the scatter graph.(1)
(Total for Question 4 is 3 marks)
5
Using the scatter graph you completed in Question 4, write down the monthly revenue for the company with 7 employees.
(Total for Question 5 is 1 mark)
6
State one feature of a good line of best fit drawn on a scatter graph.
(Total for Question 6 is 1 mark)
7
A gardener records the height of a plant each week. Number of weeks (x): 1, 2, 3, 4, 5, 6, 7, 8. Height of plant in cm (y): 5, 9, 14, 17, 22, 26, 29, 34. The points for weeks 1 to 6 have already been plotted on the grid.
(a)Plot the remaining two points from the table on the grid.(2)
(b)State the type of correlation shown.(1)
(Total for Question 7 is 3 marks)
8
Using the scatter graph from Question 7, describe what the correlation shows about the relationship between the number of weeks and the height of the plant.
(Total for Question 8 is 2 marks)
9
Nine students record how many hours they revised and their test score (%). Hours revised (x): 1, 2, 3, 4, 5, 6, 6, 7, 8. Test score % (y): 40, 45, 52, 58, 20, 68, 72, 78, 82.
(a)Write down the coordinates of the outlier.(1)
(b)Suggest a possible reason for this outlier.(1)
(Total for Question 9 is 2 marks)
10
A scatter graph shows the relationship between the number of pages in a magazine (between 20 and 60 pages) and its weight in grams. Bilal wants to use the line of best fit to estimate the weight of a magazine with 200 pages. Explain why this estimate would not be reliable.
(Total for Question 10 is 2 marks)
11
Six students recorded the number of hours they revised and their test score (%). Hours revised (x): 2, 4, 5, 7, 8, 10. Test score % (y): 45, 55, 60, 68, 75, 85.
(a)Calculate the mean number of hours revised.(1)
(b)Calculate the mean test score.(1)
(c)Write down the coordinates of the point that must lie on the line of best fit.(1)
(Total for Question 11 is 3 marks)
12
Graph A shows the relationship between arm span and height for a class of students; the points lie close to a straight line. Graph B shows the relationship between shoe size and exam score for the same class; the points are scattered with no clear pattern.
(a)State which graph shows the stronger correlation.(1)
(b)Give a reason for your answer.(1)
(Total for Question 12 is 2 marks)
13
For each statement, write True or False.
(a)A scatter graph showing a strong positive correlation between two variables proves that one variable causes the other to increase.(1)
(b)It is possible for two variables to show a strong correlation even though there is no causal link between them.(1)
(c)A line of best fit can be used to make a reliable estimate for any value of x, no matter how large.(1)
(Total for Question 13 is 3 marks)
14
A cafe owner records the temperature and the number of ice creams sold. Temperature in degrees C (x): 10, 12, 14, 16, 18, 20, 22, 24, 27, 30. Number of ice creams sold (y): 28, 38, 45, 55, 60, 68, 78, 90, 98, 112. The points for the first 8 temperatures (10 to 24 degrees C) have already been plotted on the grid.
(a)Plot the remaining two points from the table on the grid.(2)
(b)Describe the correlation shown.(1)
(c)A line of best fit has been drawn passing through (10, 30) and (30, 110). Use this line to estimate the number of ice creams sold when the temperature is 20 degrees C.(2)
(d)One day the temperature was 38 degrees C. Explain why it would not be appropriate to use the line of best fit to estimate the number of ice creams sold on this day.(1)
(Total for Question 14 is 6 marks)
15
A garage records the age and value of used cars of the same model. Age in years (x): 1, 2, 3, 4, 5, 6, 7, 8, 9. Value in pounds (y): 9300, 8700, 7900, 7300, 6400, 5900, 5000, 4500, 3700. The points for ages 1 to 7 have already been plotted on the grid.
(a)Plot the remaining two points from the table on the grid.(2)
(b)A line of best fit has been drawn passing through (0, 10000) and (10, 3000). Use this line to estimate the value of a car that is 6 years old.(2)
(c)Explain why it would not be appropriate to use this line of best fit to predict the value of the car when it is 25 years old.(2)
(d)State what the gradient of the line of best fit represents in this context.(1)
(Total for Question 15 is 7 marks)
16
A scientist plots the reaction time (in seconds) of runners against their age (in years), for ages 10 to 80. The points curve upwards steeply for ages under 20 and over 60, but are roughly flat for ages 20 to 60. Explain why it would not be appropriate to draw a single straight line of best fit for this data.
(Total for Question 16 is 2 marks)
17
A courier company records parcel weight and delivery cost. A line of best fit has been drawn passing through (2, 4) and (10, 20), where x is weight in kg and y is cost in pounds.
(a)Use the line of best fit to estimate the cost of delivering a parcel weighing 6 kg.(2)
(b)The company's official price list charges a flat fee of £2 plus £1.60 per kg. Calculate the cost for a 6 kg parcel using the price list, and compare this with your answer to part (a).(2)
(Total for Question 17 is 4 marks)
18
A teacher recorded the number of hours 12 students spent revising and their score (%) in a maths test. The scatter graph shows strong positive correlation, and the line of best fit passes through (0, 30) and (10, 90).
(a)Use the line of best fit to estimate the score of a student who revised for 12 hours, and explain why this estimate may not be reliable.(3)
(b)The teacher claims 'revising for longer causes a higher test score'. Comment on whether the scatter graph supports this claim.(2)
(Total for Question 18 is 5 marks)
19
A builder tests samples of concrete and records the number of curing days and the compressive strength (MPa). A line of best fit passes through (2, 10) and (10, 26).
(a)Use the line of best fit to estimate the compressive strength after 6 days of curing.(2)
(b)Explain why it would not be sensible to use this line of best fit to estimate the compressive strength on day 0 (before any curing has taken place).(2)
(Total for Question 19 is 4 marks)
20
A personal trainer records the height (cm) and weight (kg) of 8 gym members. Height in cm (x): 160, 165, 168, 170, 172, 175, 178, 190. Weight in kg (y): 55, 60, 63, 66, 68, 70, 73, 64.
(a)Write down the coordinates of the outlier and suggest a possible reason why this data point does not fit the trend.(2)
(b)Calculate the mean height and the mean weight of the remaining 7 members, excluding the outlier.(2)
(c)Explain how removing the outlier would affect the line of best fit.(1)
(Total for Question 20 is 5 marks)
Mark scheme · 4.20 Scatter Graphs
Question 1
B1 correct option B
Answer: B
Question 2
B1 negative correlation (oe)
Answer: Negative correlation
Question 3
(a) B1 positive correlation
(a) Answer: Positive correlation
(b) B1 negative correlation
(b) Answer: Negative correlation
(c) B1 no correlation
(c) Answer: No correlation
Question 4
(a) B2 all 10 points plotted correctly (allow small tolerance); award B1 for at least 7 points plotted correctly
B1 any correct feature, e.g. it follows the trend of the points with roughly equal numbers of points above and below it (oe), or it passes through the mean point
Answer: It should follow the trend of the data with roughly equal numbers of points above and below the line.
Question 7
(a) B1 point (7, 29) plotted correctly
(a) B1 point (8, 34) plotted correctly
(a) Answer: (7, 29) and (8, 34)
(b) B1 positive correlation
(b) Answer: Positive correlation
Question 8
B1 identifies positive correlation
B1 correct statement in context, e.g. as the number of weeks increases, the height of the plant tends to increase
Answer: There is positive correlation: as the number of weeks increases, the height of the plant tends to increase.
Question 9
(a) B1 (5, 20) cao
(a) Answer: (5, 20)
(b) B1 any sensible reason, e.g. the student may have been unwell on the day of the test, or the score may have been recorded incorrectly (oe)
(b) Answer: The student may have been unwell or unable to prepare properly on that occasion.
Question 10
B1 200 pages is outside the range of the data collected (extrapolation)
B1 the trend seen for 20 to 60 pages may not continue for much larger magazines (e.g. different paper or cover may be used)
Answer: The estimate is unreliable because 200 pages is far outside the data collected (20 to 60 pages), so it involves extrapolation and the same trend may not continue.
Question 11
(a) B1 6 cao
(a) Answer: 6 hours
(b) B1 64.7 (or 64.67, or 64 2/3) awrt 64.7
(b) Answer: 64.7% (1 d.p.)
(c) B1 (6, 64.7) ft from parts (a) and (b)
(c) Answer: (6, 64.7)
Question 12
(a) B1 Graph A
(a) Answer: Graph A
(b) B1 the points in Graph A lie closer to a straight line / are less scattered than in Graph B
(b) Answer: The points in Graph A lie closer to a straight line, showing a stronger correlation than the scattered points in Graph B.
Question 13
(a) B1 False
(a) Answer: False
(b) B1 True
(b) Answer: True
(c) B1 False
(c) Answer: False
Question 14
(a) B1 point (27, 98) plotted correctly
(a) B1 point (30, 112) plotted correctly
(a) Answer: (27, 98) and (30, 112)
(b) B1 positive correlation
(b) Answer: Positive correlation
(c) M1 correct method, e.g. gradient = (110-30)/(30-10) = 4, then 30 + 4 x (20-10)
(c) A1 70 cao
(c) Answer: 70 ice creams
(d) B1 38 is outside the range of the data (maximum recorded was 30), so this would be extrapolation and the trend may not continue
(d) Answer: 38 degrees C is outside the range of the data collected (up to 30 degrees C), so using the line here is extrapolation and unreliable.
Question 15
(a) B1 point (8, 4500) plotted correctly
(a) B1 point (9, 3700) plotted correctly
(a) Answer: (8, 4500) and (9, 3700)
(b) M1 correct method, e.g. gradient = (3000-10000)/10 = -700, then 10000 - 700 x 6
(b) A1 £5800 cao
(b) Answer: £5800
(c) B1 25 years is outside the range of the data collected (up to 9 years), so this is extrapolation
(c) B1 at 25 years the line would give a negative value (10000 - 700x25 = -7500), which is impossible, showing the linear trend cannot continue
(c) Answer: 25 years is far outside the data range, and substituting gives a negative value (-£7500), which is impossible, so the line of best fit cannot be used here.
(d) B1 the value of the car decreases by about £700 for each extra year of age
(d) Answer: The value of the car decreases by approximately £700 for each additional year of age.
Question 16
B1 the relationship is not linear (it curves / changes shape across the range)
B1 a single straight line would not model the curved trend well, giving poor estimates for the youngest and oldest runners
Answer: The data does not follow a linear pattern (it curves at the youngest and oldest ages), so a straight line of best fit would not fit the trend well and would give poor estimates at the extremes of age.
Question 17
(a) M1 correct method, e.g. gradient = (20-4)/(10-2) = 2, then 4 + 2 x (6-2)
(a) A1 £12 cao
(a) Answer: £12
(b) M1 2 + 1.60 x 6
(b) A1 £11.60, with a valid comparison, e.g. this is close to the £12 estimate from the scatter graph, showing the line of best fit gives a reasonable estimate
(b) Answer: £11.60, which is close to the £12 estimate from part (a) (a difference of only 40p).
Question 18
(a) M1 correct method, e.g. gradient = (90-30)/10 = 6, then 30 + 6 x 12
(a) A1 102 (or over 100)
(a) B1 explains estimate is unreliable, e.g. 12 hours is outside the range of the data (extrapolation) and/or a test score cannot exceed 100%
(a) Answer: 102%, which is impossible since a test score cannot exceed 100%; 12 hours is also outside the range of the data collected, so this is extrapolation and unreliable.
(b) B1 correlation does not necessarily mean causation
(b) B1 gives another possible factor that could affect scores, e.g. natural ability, quality of revision, or prior knowledge
(b) Answer: The scatter graph only shows correlation, not causation; other factors, such as natural ability or the quality of revision, could also affect test scores.
Question 19
(a) M1 correct method, e.g. gradient = (26-10)/(10-2) = 2, then 10 + 2 x (6-2)
(a) A1 18 MPa cao
(a) Answer: 18 MPa
(b) B1 day 0 is outside the range of the data collected (curing days from 2 to 10), so this is extrapolation
(b) B1 concrete needs time to cure, so there is no real evidence the same linear pattern applies right at the start of the process
(b) Answer: Day 0 is outside the tested range of curing days (2 to 10), so this is extrapolation, and there is no evidence the linear trend holds before curing has properly begun.
Question 20
(a) B1 (190, 64) cao
(a) B1 any sensible reason, e.g. the weight may have been recorded or measured incorrectly for this member (oe)
(a) Answer: (190, 64); the weight for this member may have been recorded incorrectly, since it does not fit the increasing trend.
(b) B1 mean height = 169.7 cm (awrt, from 1188/7)
(b) B1 mean weight = 65 kg cao (from 455/7)
(b) Answer: Mean height = 169.7 cm (1 d.p.), mean weight = 65 kg
(c) B1 the line of best fit would fit the remaining points more closely, better reflecting the true positive trend, since it would no longer be pulled off course by the outlier
(c) Answer: Without the outlier, the line of best fit would follow the true positive trend more closely, as it would no longer be distorted by the anomalous point.