Scatter Graphs and Line of Best Fit - Worksheets, Questions and Revision

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GCSE · Statistics

S31 Scatter Graphs and Line of Best Fit

EDEXCEL 1ST0 · Calculator allowed · about 75 minutes

Total Marks

Name: _______________________________ Date: ____ / ____ / ______

Answer ALL questions. Show all your working.

Three students each carried out a small survey and plotted their results on a scatter graph.

(a)Write down the type of correlation shown in Study A.(1)

(b)Write down the type of correlation shown in Study B.(1)

(c)Write down the type of correlation shown in Study C.(1)

(Total for Question 1 is 3 marks)

For each statement about scatter graphs, write down whether it is True or False.

(i)A scatter graph is used to display the relationship between two numerical (quantitative) variables.(1)

(ii)A line of best fit should always be drawn so that it passes exactly through every single point on a scatter graph.(1)

(iii)The points on a scatter graph should not be joined up from one point to the next with straight lines.(1)

(Total for Question 2 is 3 marks)

A cafe owner records the outdoor temperature and the number of cups of hot chocolate sold on 8 different days.

Outdoor temperature (degrees C)	2	5	7	9	12	14	16	18
Cups of hot chocolate sold	42	38	32	28	20	16	10	6

On the grid provided, plot a scatter graph to show this information.

(Total for Question 3 is 4 marks)

A student wants to draw a scatter graph to show the relationship between the age of 15 dogs (from 1 to 14 years) and their resting heart rate (from 70 to 130 beats per minute).

A) Horizontal axis 0 to 14 in steps of 2; vertical axis 60 to 140 in steps of 20, both axes numbered
B) Horizontal axis 0 to 100 in steps of 10; vertical axis 0 to 10 in steps of 1
C) Horizontal axis starting at 1 with an uneven, unnumbered scale; vertical axis 0 to 140 in steps of 20
D) Horizontal axis 0 to 14 in steps of 2; vertical axis starting at 65 with no numbers marked on the axis

(a)Write down the letter of the axis plan that would be most suitable for this scatter graph.(1)

(b)Give one reason why plan C would not be suitable.(1)

(Total for Question 4 is 2 marks)

The scatter graph shows, for 9 students, the number of hours spent revising for a test and the score they achieved.

(a)Describe the type and strength of correlation shown by the scatter graph.(1)

(b)Write down the test score of the student who revised for 3 hours.(1)

(c)Work out how many of the 9 students revised for more than 5 hours and scored more than 70%.(2)

(Total for Question 5 is 4 marks)

The scatter graph shows, for 8 used cars of the same model, the age of the car and its current value.

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

(b)Interpret what this scatter graph shows about the relationship between the age of a car and its value.(1)

(c)Explain whether it would be sensible to use the scatter graph to estimate the value of a 7-year-old car of this model.(1)

(Total for Question 6 is 3 marks)

The scatter graph shows, for 8 adults, the number of hours of exercise taken per week and their resting heart rate.

(a)Write down the coordinates of the point that does not fit the pattern of the other points.(1)

(b)Suggest one possible reason for this outlier.(1)

(c)Explain the effect that including this point would have on a line of best fit, and suggest how it should be treated.(1)

(Total for Question 7 is 3 marks)

The scatter graph shows, for 8 days at a seaside town, the value of ice cream sales and the number of sunburn cases reported that week.

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

(b)A local newspaper reports: 'Eating more ice cream causes sunburn.' Explain why this conclusion is not necessarily correct.(2)

(c)Suggest another factor that could explain the pattern shown on the scatter graph.(1)

(Total for Question 8 is 4 marks)

The scatter graph shows, for 8 cars, the engine size and the fuel used per 100 km driven. The points are plotted but no line of best fit has been drawn.

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

(b)On the grid, draw a line of best fit for this data.(2)

(c)Use your line of best fit to estimate the fuel used per 100 km for a car with an engine size of 1.8 litres.(2)

(d)Explain why it would not be appropriate to use your line of best fit to predict the fuel used per 100 km for a car with a 6.0 litre engine.(2)

(Total for Question 9 is 7 marks)

A student wants to investigate whether there is a relationship between the time Year 11 students spend on phone apps each day and the amount of sleep they get each night.

(a)Plan: Write down a suitable hypothesis for this investigation.(1)

(b)Collect: The school has 220 Year 11 students split across 8 tutor groups of different sizes. Name a sampling method the student could use to choose 20 students for the survey, and give a reason for your choice.(2)

(c)Process: The table shows the results for two more students.

Student	D	E
Phone use (hours)	6	7
Sleep (hours)	5	4

Plot the points for students D and E on the grid.(2)

(d)Interpret: The completed scatter graph shows a negative correlation. Write a conclusion for the investigation, including a comment on causation.(2)

(Total for Question 10 is 7 marks)

A gardener records the height of a sunflower plant every week for 20 weeks. The equation of the line of best fit for the data is h = 1.5w + 8, where w is the number of weeks since planting and h is the height of the plant in cm.

(a)Use the equation to estimate the height of the plant after 14 weeks.(2)

(b)Interpret what the gradient (1.5) and the intercept (8) tell you about the plant's growth.(2)

(c)Explain why it would not be sensible to use this equation to estimate the height of the plant after 100 weeks.(1)

(Total for Question 11 is 5 marks)

The scatter graph shows, for 7 houses, the distance from the town centre and the house price, together with a line of best fit that passes through the points (0, 280) and (8, 120).

(a)Find the gradient of the line of best fit, using the points (0, 280) and (8, 120).(2)

(b)Hence write down the equation of the line of best fit in the form p = md + c, where d is the distance from the town centre and p is the house price in GBP thousands.(1)

(c)Use the equation to estimate the house price for a home 5 km from the town centre.(1)

(Total for Question 12 is 4 marks)

A personal trainer records, for 8 gym members, the number of gym sessions attended per week and their monthly income.

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

(b)The trainer claims: 'This proves that going to the gym more often causes people to earn more money.' Explain why this conclusion is not necessarily correct, and suggest one other factor that could explain the pattern.(2)

(c)The trainer extrapolates the line of best fit to claim that someone attending 20 sessions a week would earn about GBP11,000 a month. Explain why this estimate is unreliable.(2)

(Total for Question 13 is 5 marks)

The scatter graph shows, for 8 players in an ice hockey team, the number of minutes of ice time in a match and the number of goals scored so far this season.

(a)Identify the coordinates of the outlier, and suggest a possible reason for it.(2)

(b)Explain whether it would be sensible to exclude this point before drawing a line of best fit for the rest of the team.(1)

(c)Using a line of best fit drawn for the remaining 7 (non-outlier) points, estimate the number of goals scored by a player with 16 minutes of ice time.(2)

(Total for Question 14 is 5 marks)

The scatter graph shows, for 6 students, the number of study hours per week and the exam score achieved, together with a line of best fit.

(a)Identify which plotted point lies furthest from the line of best fit, giving its coordinates.(1)

(b)State what this tells you about that student's result compared with the general trend.(1)

(c)Explain how the position of this point might affect confidence in predictions made from the line of best fit for students who study around 2 hours per week.(2)

(Total for Question 15 is 4 marks)

Agricultural researchers plot, for 8 farms, the monthly rainfall and the crop yield achieved, together with a line of best fit with equation y = 0.005x + 1.2, where x is the rainfall in mm and y is the crop yield in tonnes per hectare.

(a)Two more farms were surveyed.

Farm	I	J
Rainfall (mm)	350	650
Crop yield (tonnes/hectare)	2.9	3.0

Plot the points for Farm I and Farm J on the grid.(2)

(b)One of these two new points lies much further from the line of best fit than the other. Identify which farm this is, and suggest a possible reason.(2)

(c)Using the equation of the line of best fit, estimate the crop yield for a farm with 450 mm of rainfall, and comment on whether this estimate is likely to be reliable.(3)

(Total for Question 16 is 7 marks)

Mark scheme · S31 Scatter Graphs and Line of Best Fit

Question 1

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

Question 2

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

Question 3

B1 at least 3 of the 8 points plotted correctly (± half a small square)
B1 at least 6 of the 8 points plotted correctly
B1 all 8 points plotted correctly
B1 points left unjoined, with no line of best fit drawn
Answer: Scatter graph with points at (2,42), (5,38), (7,32), (9,28), (12,20), (14,16), (16,10), (18,6).

Question 4

(a) B1 A cao
(a) Answer: A
(b) B1 identifies that the horizontal scale is uneven and/or unnumbered, oe (e.g. this would make points impossible to plot or read accurately)
(b) Answer: The horizontal axis has an uneven scale with no numbers, so points could not be plotted or read accurately.

Question 5

(a) B1 strong (or fairly strong) positive correlation, oe
(a) Answer: Strong positive correlation
(b) B1 50 cao
(b) Answer: 50
(c) M1 identifies the points with x > 5, i.e. (6,62), (7,78), (8,85), (9,90)
(c) A1 3 cao
(c) Answer: 3

Question 6

(a) B1 Negative correlation cao
(a) Answer: Negative correlation
(b) B1 correct context statement, e.g. as a car gets older, its value tends to fall/decrease, oe
(b) Answer: As a car gets older, its value tends to decrease.
(c) B1 yes, with a valid reason, e.g. 7 years lies within the range of ages in the data (1 to 10 years), so this is interpolation and is reasonably reliable
(c) Answer: Yes - 7 years is within the range of the data (1 to 10 years), so this would be interpolation and a reasonably reliable estimate.

Question 7

(a) B1 (9, 85) cao
(a) Answer: (9, 85)
(b) B1 any plausible context reason, e.g. this adult may have been unwell, stressed or measured straight after a meal, so their heart rate was unusually high that day, oe
(b) Answer: The adult may have been unwell or measured under unusual conditions, giving an unusually high heart rate despite exercising a lot.
(c) B1 including it would pull the line away from the trend of the other 7 points, so it should usually be ignored/excluded when drawing the line of best fit, oe
(c) Answer: Including it would drag the line of best fit away from the true trend, so it should usually be ignored when the line is drawn.

Question 8

(a) B1 strong positive correlation, oe
(a) Answer: Strong positive correlation
(b) B1 states that correlation does not mean causation, oe
(b) B1 correct supporting explanation, e.g. both ice cream sales and sunburn cases could be caused by something else rather than one causing the other, oe
(b) Answer: Correlation does not mean causation - both variables could be affected by a third factor rather than one directly causing the other.
(c) B1 any valid confounding variable, e.g. hot, sunny weather (leads to both more ice cream sales and more sunburn), oe
(c) Answer: Hot, sunny weather - it would lead to both more ice cream sales and more sunburn, without one causing the other.

Question 9

(a) B1 Positive correlation cao
(a) Answer: Positive correlation
(b) B1 a single ruled straight line drawn across the full range of the plotted points
(b) B1 line follows the trend of the data with a roughly equal number of points on each side, e.g. close to a line from about (0.8,5) to (3.2,12)
(b) Answer: A ruled line running from approximately (0.8,5) to (3.2,12).
(c) M1 reads a value from a single ruled line of best fit at x = 1.8
(c) A1 answer consistent with their line; expected value approximately 8 litres (accept 7 to 9), ft
(c) Answer: Approximately 8 litres (accept 7-9)
(d) B1 identifies that 6.0 litres is well outside the range of the original data (0.8 to 3.2 litres), so this would be extrapolation
(d) B1 correct conclusion, e.g. the relationship may not stay linear for such a large engine, so the estimate would be unreliable, oe
(d) Answer: Unreliable - 6.0 litres is far outside the 0.8 to 3.2 litre range of the data, so this is extrapolation and the same linear pattern may not continue.

Question 10

(a) B1 a testable statement linking the two variables, e.g. students who spend more time on phone apps tend to get less sleep, oe
(a) Answer: Students who spend more time on phone apps each day tend to get less sleep at night.
(b) B1 Stratified sampling (by tutor group), oe
(b) B1 valid reason, e.g. ensures each tutor group is represented in proportion to its size, giving a more representative sample, oe
(b) Answer: Stratified sampling by tutor group, because it ensures each group is represented in proportion to its size, giving a more representative sample.
(c) B1 point for D plotted correctly at (6,5) (± half a small square)
(c) B1 point for E plotted correctly at (7,4) (± half a small square)
(c) Answer: Points plotted at (6,5) for D and (7,4) for E.
(d) B1 correct context conclusion, e.g. students who spend more time on phone apps tend to get less sleep, oe
(d) B1 caution about causation, e.g. this does not prove phone use causes less sleep - another factor, such as staying up late generally, could affect both, oe
(d) Answer: Students who spend longer on phone apps tend to sleep less, but this correlation does not prove phone use causes less sleep - another factor, such as generally staying up later, could affect both.

Question 11

(a) M1 substitutes w = 14 into h = 1.5w + 8, oe
(a) A1 29 cm cao
(a) Answer: 29 cm
(b) B1 gradient: the plant grows by about 1.5 cm each week, oe
(b) B1 intercept: the plant's height at the time of planting (w = 0) was about 8 cm, oe
(b) Answer: The gradient shows the plant grows about 1.5 cm per week; the intercept shows the plant was about 8 cm tall when it was planted.
(c) B1 100 weeks is far outside the 20-week range of the data (extrapolation), so the plant's growth pattern may no longer be linear (e.g. it may have stopped growing or died), oe
(c) Answer: 100 weeks is far beyond the 20-week study, so this would be extrapolation and the plant's growth would very likely no longer follow the same straight-line pattern.

Question 12

(a) M1 gradient = (120 - 280) / (8 - 0), oe
(a) A1 -20 cao
(a) Answer: -20
(b) B1 p = -20d + 280 cao, ft from (a)
(b) Answer: p = -20d + 280
(c) B1 180 (GBP thousands) cao, ft from (b)
(c) Answer: GBP180,000

Question 13

(a) B1 positive correlation, oe
(a) Answer: Positive correlation
(b) B1 states that correlation does not prove causation, oe
(b) B1 valid confounding factor, e.g. people with flexible, well-paid jobs may have more free time and money to spend on both gym sessions and training, oe
(b) Answer: Correlation does not prove causation. People with well-paid, flexible jobs may simply have more free time and money to spend on the gym, rather than the gym causing higher income.
(c) B1 identifies that 20 sessions is far outside the range of the data (1 to 8 sessions), so this is extrapolation
(c) B1 correct supporting reason, e.g. the pattern may not continue at such extreme values, and attending 20 sessions a week is not realistically possible, oe
(c) Answer: Unreliable - 20 sessions a week is far outside the 1 to 8 session range of the data, so this is extrapolation, and the linear pattern (and the claim itself) is unlikely to hold at such an extreme value.

Question 14

(a) B1 (8, 17) cao
(a) B1 plausible reason, e.g. this player may be an unusually prolific scorer who makes the most of limited ice time, or benefited from lucky rebounds, oe
(a) Answer: (8, 17) - possibly an unusually prolific scorer, or a run of fortunate goals in limited playing time.
(b) B1 yes, with valid reason, e.g. keeping it would distort the line away from the trend followed by the other 7 players, oe
(b) Answer: Yes - keeping it in would pull the line of best fit away from the trend shown by the other 7 players.
(c) M1 reads a value from a sensible line of best fit through the 7 remaining points at x = 16, ignoring (8,17)
(c) A1 answer consistent with their line; expected value approximately 7.5 (accept 7 to 8), ft
(c) Answer: Approximately 7-8 goals

Question 15

(a) B1 (2, 25) cao
(a) Answer: (2, 25)
(b) B1 this student scored much lower than the trend for their study hours would predict, oe
(b) Answer: This student scored much lower than would be expected for the amount of time they spent studying.
(c) B1 identifies that there is more scatter/variation around the line at low study hours than elsewhere, oe
(c) B1 correct conclusion, e.g. predictions for students who study around 2 hours are less reliable than predictions elsewhere on the line, oe
(c) Answer: There is more variation around the line at low study hours, so predictions made for around 2 hours of study are less reliable than predictions made elsewhere on the line.

Question 16

(a) B1 Farm I plotted correctly at (350, 2.9) (± half a small square)
(a) B1 Farm J plotted correctly at (650, 3.0) (± half a small square)
(a) Answer: Points plotted at (350, 2.9) for Farm I and (650, 3.0) for Farm J.
(b) B1 Farm J cao
(b) B1 plausible reason, e.g. a local problem such as drought, pests or poor soil reduced Farm J's yield despite average rainfall, oe
(b) Answer: Farm J - a local factor such as drought, pests or poor soil may have reduced its yield despite roughly average rainfall.
(c) M1 substitutes x = 450 into y = 0.005x + 1.2, oe
(c) A1 3.45 tonnes/hectare cao (accept awrt 3.4-3.5)
(c) B1 valid comment, e.g. 450 mm is within the 100-800 mm range of the original data, so this is interpolation and should be reasonably reliable, though Farm J shows individual farms can still vary from the line, oe
(c) Answer: 3.45 tonnes/hectare - reasonably reliable, since 450 mm is within the 100-800 mm range of the data (interpolation), although Farm J shows individual results can still vary.

Question 16

7 marks

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