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H11 Regression and the PMCC

EDEXCEL 1ST0 · Calculator allowed · about 90 minutes

Total Marks

Name: _______________________________ Date: ____ / ____ / ______

Answer ALL questions. Show all your working.

Each of the following scatter graphs shows paired data collected from a different study.

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

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

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

(Total for Question 1 is 3 marks)

Which of these values could not be a product moment correlation coefficient (PMCC)?

A) 0.85
B) -0.42
C) 1.20
D) 0

(Total for Question 2 is 1 mark)

Match each statistical term to its correct definition.

Definitions:
P: A numerical measure, between -1 and 1, of the strength and direction of the linear relationship between two variables.
Q: Using a line of best fit to estimate a value within the range of the given data.
R: Using a line of best fit to estimate a value outside the range of the given data.
S: A straight line drawn through a scatter graph that best represents the linear trend in the data.

(i)Interpolation(1)

(ii)Extrapolation(1)

(iii)Line of best fit(1)

(iv)Product moment correlation coefficient (PMCC)(1)

(Total for Question 3 is 4 marks)

A gardener records, for 7 tomato plants, the number of hours of direct sunlight each plant receives per day and the number of tomatoes it produces that season. The scatter graph shows the results.

(a)Write down the number of tomatoes produced by the plant that received 5 hours of sunlight per day.(1)

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

(c)Describe what the scatter graph shows about the relationship between hours of sunlight and the number of tomatoes produced.(2)

(Total for Question 4 is 4 marks)

Four studies calculate the product moment correlation coefficient, r, between two variables:

A: r = 0.89
B: r = -0.97
C: r = 0.15
D: r = -0.42

Match each description below to the correct study.

(i)A very strong negative correlation.(1)

(ii)A strong positive correlation.(1)

(iii)A weak (or negligible) correlation.(1)

(iv)A moderate negative correlation.(1)

(Total for Question 5 is 4 marks)

For each statement about the PMCC and regression lines, write down whether it is True or False.

(i)The value of the product moment correlation coefficient, r, must always lie between -1 and 1 inclusive.(1)

(ii)A product moment correlation coefficient of r = 0 always means there is no relationship at all between the two variables.(1)

(iii)The regression line of y on x always passes through the point (x-bar, y-bar), the means of the x and y data.(1)

(iv)If two variables have a strong linear correlation, this always means that changes in one variable directly cause changes in the other.(1)

(Total for Question 6 is 4 marks)

A personal trainer records, for 8 gym members, the number of personal training sessions they have attended and the increase in the weight (kg) they can bench press since starting. The scatter graph shows the results, with the points plotted but no line of best fit drawn.

(a)State one feature, other than being drawn with a ruler, that a well-drawn line of best fit should have.(1)

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

(c)Use your line of best fit to estimate the increase in bench press weight after 4.5 sessions.(2)

(d)State whether your estimate in part (c) was found by interpolation or extrapolation.(1)

(Total for Question 7 is 6 marks)

For a set of paired data, the regression line of y on x is y = 1.5x + 8. The mean of the x-values is x-bar = 6 and the mean of the y-values is y-bar = 17.

(a)Show that the point (x-bar, y-bar) lies on this regression line.(2)

(b)Explain why this result is not a coincidence.(1)

(Total for Question 8 is 3 marks)

For a sample of 12 days, a museum manager records the number of visitors and the average time (minutes) spent in the gift shop. She calculates Sxy = 245, Sxx = 410 and Syy = 198.

You may use: r = Sxy / √Sxx * Syy.

(a)Calculate the value of r, giving your answer correct to 3 significant figures.(3)

(b)Interpret the value of r in the context of the number of visitors and time spent in the gift shop.(2)

(Total for Question 9 is 5 marks)

A delivery company records, for 7 journeys, the distance travelled (miles) and the fuel used (litres). The scatter graph shows the results, together with a line of best fit that passes through the points (20, 5) and (80, 17).

(a)Find the equation of the line of best fit in the form y = mx + c, where x is the distance travelled and y is the fuel used.(3)

(b)Use the equation to estimate the fuel used for a distance of 50 miles.(1)

(c)The distances in the sample ranged from 20 to 80 miles. Comment on the reliability of using this equation to estimate the fuel used for a 150-mile journey.(2)

(Total for Question 10 is 6 marks)

For a sample of 10 runners whose weekly training distance ranged from 10 to 40 miles, the regression line relating weekly training distance, d (miles), to 10 km race time, T (minutes), is:

T = 65 - 0.8d

(a)Use the equation to estimate the race time for a runner who trains 25 miles per week.(2)

(b)Explain why it would not be appropriate to use this equation to estimate the race time for a runner who trains 80 miles per week.(2)

(c)Interpret what the value -0.8 represents in this context.(1)

(d)State whether the value 65 (the intercept) has a sensible real-world interpretation in this context, and explain your answer.(2)

(Total for Question 11 is 7 marks)

A researcher plots data for two variables, x and y. The scatter graph shows a clear curved (U-shaped) pattern. She calculates the product moment correlation coefficient and finds r = 0.08, and concludes that 'there is no relationship at all between x and y'.

(a)State what a value of r close to 0 normally suggests about the linear relationship between two variables.(1)

(b)Explain why the researcher's conclusion that 'there is no relationship between x and y' is not fully justified, given the shape of the scatter graph.(2)

(Total for Question 12 is 3 marks)

A student wants to find out whether there is a relationship between the number of hours Year 11 students spend using a revision app each week and their score in a mock maths exam. She plans her investigation using the four stages of the statistical enquiry cycle: plan, collect, process and interpret.

(a)(Plan) Explain why she needs to record app-usage hours and mock exam score as paired data from the same students, rather than collecting each separately from different groups.(1)

(b)(Collect) She only surveys students in the top set for maths. Give one reason why this could make her results unreliable for Year 11 as a whole.(1)

(c)(Process) For her sample of 20 students, she calculates Sxy = 180, Sxx = 50 and Syy = 900. Calculate the value of r, giving your answer correct to 3 significant figures.(3)

(d)(Interpret) Interpret this value of r, and explain why a strong correlation would not prove that using the revision app causes higher exam scores.(2)

(Total for Question 13 is 7 marks)

A researcher studies the relationship between the number of hours of extra tuition a student receives per week, x, and their percentage score, y, in a mock exam. For a sample of 8 students, whose weekly tuition hours ranged from 2 to 9, she calculates Sxy = 96 and Sxx = 40. The mean number of tuition hours is x-bar = 5 and the mean percentage score is y-bar = 62.

You may use: gradient b = Sxy / Sxx, and a = y-bar - b(x-bar), for the regression line y = a + bx.

(a)Calculate the gradient, b, of the regression line of y on x.(2)

(b)Hence find the equation of the regression line in the form y = a + bx.(2)

(c)Interpret the value of the gradient in context.(2)

(d)Use the equation to predict the score of a student who receives 6 hours of tuition per week.(2)

(e)State, with a reason, whether this estimate is likely to be reliable.(1)

(Total for Question 14 is 9 marks)

A scientist measures, for 6 plots of land, the amount of fertiliser applied (grams per square metre), x, and the yield of a crop (kg), y.

Plot	Fertiliser, x (g/m²)	Yield, y (kg)
1	10	12
2	15	18
3	20	20
4	25	25
5	30	28
6	35	33

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.(4)

(b)Hence show that the value of the product moment correlation coefficient, r, is 0.994 correct to 3 significant figures.(3)

(c)Interpret the value of r in the context of fertiliser applied and crop yield.(2)

(d)Give one reason, other than measurement error, why the correlation is very strong but not perfect (r = 1).(1)

(Total for Question 15 is 10 marks)

A student collects data for only 4 pairs of values and calculates r = 0.99. He claims that this proves a strong, reliable linear relationship exists between the two variables in general.

Evaluate this claim.

(Total for Question 16 is 4 marks)

Mark scheme · H11 Regression and the PMCC

Question 1

(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 2

B1 C cao
Answer: C) 1.20

Question 3

(i) B1 Q cao
(i) Answer: Q
(ii) B1 R cao
(ii) Answer: R
(iii) B1 S cao
(iii) Answer: S
(iv) B1 P cao
(iv) Answer: P

Question 4

(a) B1 14 cao
(a) Answer: 14
(b) B1 Positive cao
(b) Answer: Positive correlation
(c) B1 identifies that as hours of sunlight increases, the number of tomatoes produced increases, oe
(c) B1 correct context, e.g. plants that received more sunlight tended to produce more tomatoes, oe
(c) Answer: As the number of hours of sunlight increases, the number of tomatoes produced also tends to increase.

Question 5

(i) B1 B cao
(i) Answer: B
(ii) B1 A cao
(ii) Answer: A
(iii) B1 C cao
(iii) Answer: C
(iv) B1 D cao
(iv) Answer: D

Question 6

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

Question 7

(a) B1 follows the direction/trend of the points, passing through (or close to) the mean point, with a roughly equal number of points above and below the line, oe
(a) Answer: It should follow the trend of the data with roughly equal numbers of points on each side of the line.
(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 (1,2) to (8,13)
(b) Answer: A ruled line running from approximately (1,2) to (8,13).
(c) M1 reads a value from a single ruled line of best fit at x = 4.5
(c) A1 answer consistent with their line; expected value approximately 7.5 kg (accept 6.5 to 8.5), ft
(c) Answer: Approximately 7.5 kg (accept 6.5-8.5)
(d) B1 Interpolation cao
(d) Answer: Interpolation

Question 8

(a) M1 substitutes x = 6 into y = 1.5x + 8
(a) A1 y = 1.5(6) + 8 = 17, which equals y-bar, confirming the point lies on the line
(a) Answer: y = 1.5(6) + 8 = 17 = y-bar, so (6,17) lies on the line.
(b) B1 the regression line of y on x always passes through the mean point (x-bar, y-bar), so this must be true for any regression line, oe
(b) Answer: The regression line of y on x always passes through the mean point (x-bar, y-bar), so it must pass through (6,17).

Question 9

(a) M1 substitutes correctly into r = Sxy / √Sxx * Syy
(a) A1 r = 0.8599... (or equivalent unrounded value)
(a) A1 0.860 (3 sf), ft
(a) Answer: 0.860
(b) B1 identifies a strong positive correlation/association
(b) B1 correct context, e.g. days with more visitors tend to see more average time spent in the gift shop, oe
(b) Answer: There is a strong positive correlation - days with more visitors tend to see more time spent, on average, in the gift shop.

Question 10

(a) M1 gradient = (17-5)/(80-20), oe
(a) A1 m = 0.2
(a) A1 y = 0.2x + 1 cao (c found correctly, e.g. from 5 = 0.2(20) + c)
(a) Answer: y = 0.2x + 1
(b) B1 11 litres cao, ft from (a)
(b) Answer: 11 litres
(c) B1 identifies that 150 miles is well outside the range of the original data (20 to 80 miles), so this is extrapolation
(c) B1 correct conclusion, e.g. the linear pattern may not continue that far (fuel use per mile could change on longer/motorway journeys), so the estimate would be unreliable, oe
(c) Answer: Unreliable - 150 miles is far outside the 20-80 mile range of the data, so this is extrapolation and the same linear relationship may not continue.

Question 11

(a) M1 substitutes d = 25 into T = 65 - 0.8d
(a) A1 45 minutes cao
(a) Answer: 45 minutes
(b) B1 80 miles is well outside the range of the original data (10 to 40 miles), so this would be extrapolation
(b) B1 correct conclusion, e.g. the linear relationship may not continue at such high training distances (e.g. risk of injury/diminishing returns), so the estimate would be unreliable, oe
(b) Answer: 80 miles is far outside the 10-40 mile range of the data, so using the equation here is extrapolation and may give an unreliable estimate.
(c) B1 for each extra mile trained per week, race time decreases by about 0.8 minutes, on average, oe
(c) Answer: For each extra mile trained per week, race time decreases by about 0.8 minutes, on average.
(d) B1 No cao
(d) B1 d = 0 lies outside the range of the data (10 to 40 miles), so it is not appropriate to interpret it directly - a runner who does no weekly training would not be represented by this model, oe
(d) Answer: No; d = 0 lies outside the range of the data (10-40 miles), so it should not be directly interpreted - it would represent a runner with no training at all, which is not represented in the data.

Question 12

(a) B1 there is little or no LINEAR relationship between the variables, oe
(a) Answer: There is little or no linear relationship between the variables.
(b) B1 the PMCC only measures the strength of a LINEAR relationship, so a small r does not rule out a strong non-linear (e.g. curved) relationship
(b) B1 the scatter graph shows a clear curved (U-shaped) pattern, so x and y are actually strongly related, just not in a straight-line way, oe
(b) Answer: r only measures linear relationships; the scatter graph clearly shows a strong curved (U-shaped) pattern, so x and y are strongly related - just not in a straight line, which is why r is close to 0 despite the clear pattern.

Question 13

(a) B1 a correlation can only be found between two variables recorded for the same individuals, so both measurements must come from the same students, oe
(a) Answer: Correlation can only be measured between two variables recorded for the same individuals, so both values must come from the same students.
(b) B1 the top set is not representative of all Year 11 students (e.g. generally higher attainers), so the results may not apply to other students, oe
(b) Answer: Top-set students are not representative of all Year 11 students, so the relationship found may not hold for the wider year group.
(c) M1 substitutes correctly into r = Sxy / √Sxx * Syy
(c) A1 r = 0.8485... (or equivalent unrounded value)
(c) A1 0.849 (3 sf)
(c) Answer: 0.849
(d) B1 correctly interprets r = 0.849 as a strong positive correlation, e.g. students who use the app more tend to score higher
(d) B1 identifies that this does not prove causation, e.g. a confounding variable such as general motivation or overall time spent revising (not just on the app) could affect both, oe
(d) Answer: r = 0.849 shows a strong positive correlation - students who used the app more tended to score higher - but this does not prove the app causes higher scores, since a confounding factor (e.g. a generally more motivated student both uses the app more and revises more overall) could explain the link instead.

Question 14

(a) M1 96/40, oe
(a) A1 2.4 cao
(a) Answer: 2.4
(b) M1 substitutes into a = 62 - 2.4(5), oe
(b) A1 y = 2.4x + 50 cao
(b) Answer: y = 2.4x + 50
(c) B1 identifies that score increases as tuition hours increase
(c) B1 correct size/context, e.g. for each extra hour of tuition per week, percentage score increases by about 2.4 percentage points, on average, oe
(c) Answer: For each extra hour of tuition per week, a student's percentage score increases by about 2.4 percentage points, on average.
(d) M1 substitutes x = 6 into y = 2.4x + 50
(d) A1 64.4 cao
(d) Answer: 64.4%
(e) B1 reliable/likely to be reasonably accurate because 6 lies within the range of the sampled data (2 to 9 hours), so this is interpolation, oe
(e) Answer: Likely to be fairly reliable, since 6 hours lies within the range of the data collected (2 to 9 hours), so this is interpolation.

Question 15

(a) M1 finds sum x = 135, sum y = 136, sum xy = 3410, sum x² = 3475 and sum y² = 3366
(a) A1 Sxy = 350
(a) A1 Sxx = 437.5
(a) A1 Syy = 283.3 (awrt), or 1700/6
(a) Answer: Sxy = 350, Sxx = 437.5, Syy = 283.3 (1 dp)
(b) M1 r = 350 / √437.5 x 283.33..., ft from (a)
(b) A1 r = 0.9941... (unrounded)
(b) A1 0.994 (3 sf) shown, cao
(b) Answer: 0.994
(c) B1 identifies a very strong positive correlation
(c) B1 correct context, e.g. plots given more fertiliser tended to produce a higher crop yield, oe
(c) Answer: There is a very strong positive correlation - plots given more fertiliser tended to produce a higher crop yield.
(d) B1 other factors also affect crop yield, e.g. rainfall, soil quality, sunlight or pest damage, so yield is not determined by fertiliser alone, oe
(d) Answer: Other factors also affect yield, such as rainfall, soil quality or sunlight, so fertiliser is not the only thing determining crop yield.

Question 16

B1 recognises that with only 4 data pairs, a high value of r can occur easily by chance, and is not a reliable estimate of the true relationship, oe
B1 notes that a larger sample would give more confidence that the relationship is genuine, oe
B1 notes that even a very strong, reliable correlation would still not by itself prove causation or necessarily generalise beyond the data collected, oe
B1 reaches a balanced conclusion, e.g. r = 0.99 suggests a strong linear association within this small sample, but the claim of a reliable general relationship is not fully justified without more data, oe
Answer: The claim is not fully justified: with only 4 data pairs, a high r value can occur by chance and does not reliably show a genuine underlying relationship; a larger sample would be needed before concluding the relationship is strong and reliable in general.

Regression and the PMCC - Worksheets, Questions and Revision