Correlation & Regression (DP IB Maths: AI SL)

Exam Questions

4 hours25 questions

2 marks

A teacher collected the maths and physics test scores of a number of students and drew a scatter diagram to represent this data.

q1a-4-2-correlation-regression-medium-ib-ai-sl-maths-screenshot

Describe the correlation shown by the scatter diagram, and interpret the correlation in context.

How did you do?

2 marks

An alternative therapist collected data on his clients’ reported levels of anxiety as well as the number of trees they had hugged in the course of therapy. He drew a scatter diagram to represent this data.

q1a-2-4-2-correlation-regression-medium-ib-ai-sl-maths-screenshot

Describe the correlation shown by the scatter diagram, and interpret the correlation in context.

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4 marks

Jennifer sells cups of tea at her shop and has noticed that she sells more tea on cooler days. On five different days, she records the maximum daily temperature, T, measured in degrees Celsius, and the number of cups of teas sold,C The results are shown in the following table.

Maximum Daily Temperature, T.	3	5	8	9	12
Cups of tea sold,C.	37	34	33	26	21

(i)

Write down the equation of the regression line of C on T.

(iii)

Write down the value of the Pearson’s product-moment correlation coefficient, r.

How did you do?

2 marks

Use your regression equation from part (a)(i) to estimate the number of teas that Jennifer will sell on a day when the maximum temperature is 11° C.

How did you do?

2 marks

Being sure to consider the result from part (a)(ii) in your answer, state how confident you would be in your estimate from part (b).

How did you do?

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4 marks

The following table shows the mean height, y cm, of primary school children who are age x years old.

Age, x years	6.25	7.35	8.5	9.25	10.75
Mean Height, y cm	115	121	129	136	140

(i)

Write down the equation of the regression line of y on x.

(ii)

Write down the value of the Pearson’s product-moment correlation coefficient, r .

How did you do?

2 marks

Use your regression equation from part (a)(i) to estimate the height of a child aged 9 years old.

How did you do?

1 mark

Explain why it is not appropriate to use the regression equation to estimate the age of a child who is 133 cm tall.

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4 marks

Rebecca, a regular jogger, ran the “Thao Dien Loop” on 7 consecutive days. The following table shows the distance, x km, that she ran and the corresponding number of calories, y, that she was able to burn during the run.

Distance (x)	2	5	6	7	10	12	14
Calories (y)	180	315	365	435	619	830	871

The number of calories burnt during a run is dependent upon on the length of the run.

(i)

Write down the equation of the regression line of y on x, giving your answer in the form

y = a x + b

where a and b are constants to be found.

(ii)

Write down the value of the Pearson’s product-moment correlation coefficient, r.

How did you do?

1 mark

Interpret, in the context of the question, the value of a found in part (a)(i).

How did you do?

2 marks

On the 8th day, Rebecca is only able to run for 8 kilometres.

Use the result from part (a)(i) to estimate the number of calories Rebecca will lose.

How did you do?

1 mark

Comment on the validity of using the result from part (a)(i) to answer part (c).

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4 marks

The percentage of people who are willing to get a particular vaccine is dependent on their age. The following table shows the age, A years old, and the corresponding percentage of people, V. that are willing to receive a vaccine for 6 different ages.

Age, (A)	25	30	35	40	45	50
Percentage of willing people, (V)	57	59	61	62	68	75

(i)

Write down the equation of the regression line of V on A, giving your answer in the form

V = a A + b

where a and b are constants to be found.

(ii)

Write down the value of the Pearson’s product-moment correlation coefficient, r.

How did you do?

1 mark

Interpret, in the context of the question, the value of a found in part (a)(i).

How did you do?

2 marks

Use the result from part (a)(i) to estimate the percentage of people aged 95 years old who are in willing to receive a vaccine.

How did you do?

1 mark

Comment on the validity of using the result from part (a)(i) to answer part (c).

How did you do?

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4 marks

The price, $P , of an airline ticket is dependent on the distance, d km, between two cities. The table below shows the airfares in US dollars from Prague in the Czech Republic, to eight different destinations in Europe.

Distance (d)	885	340	835	330	1270	295	650	1930
Price (P)	99	50	90	45	119	42.5	59	139

The relationship between and can be modelled by the regression line of P on d with equation $P = a d + b .$

(i)

Write down the equation of the regression line of P on d

(ii)

Write down the value of the Pearson’s product-moment correlation coefficient, r

How did you do?

2 marks

Use the result from part (a) to estimate the price of an airline ticket for a flight from Prague to a destination that is 2635 km away.

How did you do?

2 marks

Madlenka buys a ticket for a flight from Prague to Cairo, a distance of 2635 km. The airfare in US dollars is $245.

Compare this price to your result from part (b), suggesting possible reasons for any discrepancies.

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3 marks

A snack company is trialling a new series of crisp flavours. The eight flavours have been assigned the letters A through H, and the company has conducted taste tests in which volunteers assign each flavour a score on a scale from 1 to 10 (where ‘1’ indicates ‘I hate it!’, and ‘10’ indicates ‘This is my new favourite!’). The following table collates the scores assigned to the flavours by each of three volunteers – Idris, Jameel and Kevin.

Flavours	A	B	C	D	E	F	G	H
Idris’ score	1	9	4	8	10	3	7	6
Jameel’s score	6	4	2	10	7	9	3	8
Kevin’s score	9	4	7	2	1	7	5	6

The company would like to find the Spearman’s rank correlation coefficients for these taste testers’ rankings.

Complete the information in the following table.

Flavours	A	B	C	D	E	F	G	H
Idris’ rank	1				8
Jameel’s rank			1	8
Kevin’s rank	8				1	6.5

How did you do?

3 marks

Find the value of the Spearman’s rank correlation coefficient, r_s, for:

(i) Idris’ and Jameel’s ranks

(ii) Idris’ and Kevin’s ranks

(iii) Jameel’s and Kevin’s ranks.

How did you do?

3 marks

Comment, in the context of the question, on the results obtained for $r_{s}$ in part (b)(i), (ii) and (iii).

How did you do?

2 marks

It is noticed that a greasy fingerprint has caused Kevin’s score of 9 for flavour A to be misread. The score actually assigned by Kevin to flavour A was 8.

Explain, with a reason, whether this will change any of the values for the Spearman’s rank correlation coefficient r_s that were calculated in part (b).

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3 marks

The following table collates the scores achieved on a recent maths test and a recent physics test by a group of 10 students.

Student	A	B	C	D	E	F	G	H	I	J
Maths score	62	79	71	56	61	84	86	76	56	11
Physics score	54	62	70	65	82	83	74	91	90	7

The scatter diagram for these scores is shown in the diagram below:

q8a-4-2-correlation-regression-medium-ib-ai-sl-maths-screenshot

Write down the value of the Pearson’s product-moment correlation coefficient, r,

(i) with student J included

(ii) without student J included.

How did you do?

2 marks

Complete the information in the following table, ranking the scores from highest to lowest:

Student	A	B	C	D	E	F	G	H	I	J
Maths rank			5	8.5			1			10
Physics rank							5	1		10

How did you do?

3 marks

Find the value of the Spearman’s rank correlation coefficient, r_s,

(i) with student J included

(ii) without student J included.

How did you do?

3 marks

Comment on the results of parts (a) and (c).

How did you do?

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2 marks

10 students are asked to give a score from 0 to 10 on how much they enjoy watching football and how much they enjoy watching cricket. The scores are shown in the scatter plot below.

q1a-4-2-correlation-regression-hard-ib-ai-sl-maths-screenshot

Use the scatter diagram to complete the missing cells in the table below.

Football score, x	1	3	4	4	5	5	7	8	10	10
Cricket score, y			4			10				2

How did you do?

2 marks

Another student only gave a cricket score of 6 and no football score.

Estimate the football score for the student who has a cricket score of 6.

How did you do?

1 mark

Comment on the reliability of your estimate found in part (b).

How did you do?

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4 marks

A study is conducted on 6 participants (labelled A to F) measuring their average number of hours of sleep per night and their score, from 0 to 100, in a short-term memory test. The results of the study are shown in the table below.

Participant	A	B	C	D	E	F
Average number of hours of sleep	6.8	7.2	8.1	9.4	5.9	7.5
Short term memory test score	72	70	82	79	62	80

Draw a scatter diagram for the above data on the axes below.

q2a-4-2-correlation-regression-hard-ib-ai-sl-maths-screenshot

How did you do?

3 marks

(i)

Calculate the Pearson’s product-moment correlation coefficient, r.

(ii)

Write down the equation of the regression line y on x.

(iii)

Draw the regression line on your scatter diagram.

How did you do?

2 marks

How did you do?

1 mark

How did you do?

Did this page help you?

3 marks

The following table shows the total CO₂ emissions, T tonnes, from 5 different countries (labelled A to E) and their average annual household income,S USD.

Country	A	B	C	D	E
CO₂emissions, T tonnes	10 500 000	5 500 000	2 500 000	1 600 000	1 200 000
Average annual household income, S USD	55 000	105 000	15 000	55 000	40 000

(i)

Calculate the Pearson’s product-moment correlation coefficient, r.

(ii)

Hence comment on the result.

How did you do?

2 marks

Write down the equation of the regression line S on T.

How did you do?

2 marks

State two reasons why it would be inappropriate to use the regression line from part (b) to estimate the percentage of total global emissions from a country where the average household annual income is 75 000 USD.

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2 marks

Sweet Dreams is a company that hotel ratings website that personally rates hotels on a scale of 1 to 10 along with allowing the public to leave ratings on the hotels. The ratings from the public include a comment section. The table below shows the ratings from Sweet dreams and the public for 9 different hotels (labelled A to I).

Hotel	A	B	C	D	E	F	G	H	I
Sweet Dream’s rating	7	9	4	5	8	1	3	6	2
Public’s rating	7.8	9.1	2.1	2.4	9.8	1.4	1.9	5.5	1.5
Sweet Dream’s rank
Public’s rank

Complete the two empty rank rows in the table above, ranking the highest scores first.

How did you do?

3 marks

(i)

Calculate the value of the Pearson’s product-moment correlation coefficient, r.

(ii)

Calculate the value of the Spearman’s rank correlation coefficient, r_s .

How did you do?

1 mark

A mistake was made when calculating the rating from the public for Hotel D. The real rating for Hotel D from the public is 2.3.

Explain why r_s does not change when the public’s rating for Hotel D is changed.

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4 marks

Easy Breezes is a company based in the snowy mountains of Greenland that makes heat pumps. Easy Breezes wants to see if the average weekly temperature, in °C, is correlated with the average weekly energy consumption from their air conditioning units, in kilowatt hours (kWH). Easy Breezes records the following data.

Average weekly temperature,

in °C(x)

-8.2

-4.3

-1.7

0.8

2.0

4.4

8.5

Average weekly energy consumption, in kWH (y)

365.2

316.4

292.7

249.1

187.2

142.8

131.2

(i)

Calculate the Pearson’s product-moment correlation coefficient, r.

(ii)

Write down the equation of the regression line y on x.

How did you do?

4 marks

(i)

Use your regression line from part (a) (i) to estimate the number of kWH one of Easy Breezes air conditioning units would use in a week when the weekly average temperature is 11° C.

(ii)

Comment on the reliability of your estimate.

How did you do?

2 marks

The actual usage of one of their heat pumps in a week where the average temperature was 11°C is 52.0 kWH.

Calculate the percentage error in your estimate found in part (b) (i) and the actual usage.

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3 marks

A supermarket has a physical and online store. The following table shows the total daily revenue, y, in USD, along with the number of customers that they had come into their physical store during the day, over 8 separate days.

Customers (x)	45	88	54	97	154	101	36	72
Revenue, USD (y)	548.21	832.55	497.71	1021.97	1138.73	988.62	1026.21	754.38

(i)

Calculate the Pearson’s product-moment correlation coefficient, r.

(ii)

Hence comment on the result.

How did you do?

3 marks

The regression line on can be written in the form $y = a + b x .$ .

Calculate the values of a and b and interpret their meanings.

How did you do?

2 marks

The supermarket has daily fixed costs of 650 USD.

Find an expression for the daily profit of the supermarket when they have x customers on a particular day.

How did you do?

2 marks

Estimate the least number of physical customers required in order to make a profit on any particular day.

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4 marks

10 rugby players (labelled A to J) are used to investigate the relationship between a player’s maximum sprint velocity, in ms^-1, and their weight, in kg. The data is recorded in the table below.

Player	A	B	C	D	E	F	G	H	I	J
Weight, in kg (x)	96	99	88	95	112	98	85	108	82	109
Maximum sprint velocity, in ms^-1 (y)	7.5	6.9	10.1	8.8	6.1	6.9	5.8	10.7	11.9	6.5

Calculate the value of the Pearson’s product-moment correlation coefficient, $r$

(i)

with players G and H.

(ii)

without the players G and H.

How did you do?

4 marks

Write down the equation of the regression line y on x

(i)

with players G and H.

(ii)

without the players G and H.

How did you do?

2 marks

Comment on the results found in part (a) and (b) and state whether you would use the regression line with players G and H or the regression line without players G and H when estimating a rugby player’s maximum sprint velocity, given their weight.

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1 mark

A study was conducted on 6 students measuring their arm length, x cm, and the maximum number of push ups they can do in a minute. The results of the study are shown in the table below.

Arm length, x cm	72.2	69.2	75.6	78.1	78.5	74.5
No. of push-ups, y	34	42	24	30	38	31

State the range of the number of push ups.

How did you do?

3 marks

(i)

Calculate the Pearson’s product-moment correlation coefficient, r.

(ii)

Comment on the correlation between the athlete’s arm length and the maximum number of push ups they can do in a minute.

How did you do?

2 marks

Write down the equation of the regression line y on x.

How did you do?

2 marks

Another student, Tom, is a sportsman and can do 62 push ups in one minute.

Use the regression line found in part (c) to estimate Tom’s arm length.

How did you do?

2 marks

State whether your estimate is valid and justify your answer.

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3 marks

Body mass index (BMI) is used to measure whether someone is over or under weight, however BMI does not take someone’s body fat percentage into account. The following table shows the BMI and body fat percentage from 7 male participants.

BMI, x	22.4	19.8	25.5	29.8	31.2	18.1	17.2
Body fat %, y	22.1	20.1	24.2	31.1	16.2	15.1	8.6

(i)

Calculate the Pearson’s product-moment correlation coefficient,

r

(ii)

Comment on the result found for r.

How did you do?

4 marks

The regression line y on x is in the form $y = m x + c$ .

Calculate the values of m and c and interpret their meanings. Explain whether they are appropriate within the context of the question.

How did you do?

3 marks

The formula to calculate someone’s BMI is

$BMI = \frac{weight in kilograms}{(height in metres)^{2}}$

John weighs 95 kilograms and is 1.84 metres tall.

Estimate John’s body fat percentage and comment on the reliability of your estimate.

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10a

4 marks

A movie cinema is considering significantly reducing the price of their popcorn as they believe their customers spend more on drinks when they buy popcorn. They recorded the following data of the daily revenue from popcorn, $x , and the daily revenue from drinks,$y over 8 randomly selected days.

Popcorn revenue, $x	78.20	102.50	30.80	22.20	132.90	200.50	154.80	132.40
Drinks revenue, $y	202.10	308.50	60.70	75.80	270.50	300.00	368.20	198.70

Calculate

(i)

\bar{x}

, the mean daily revenue from popcorn.

(ii)

\bar{y}

, the mean daily revenue from drinks.

(iii)

r, the Pearson’s product-moment correlation coefficient.

How did you do?

10b

4 marks

The equation of the regression line y on x is in the form $y = a + b x$ .

Calculate the values of a and b and interpret their meanings and explain whether they are appropriate within the context of the question.

How did you do?

10c

2 marks

Show that the point $M (\bar{x}, \bar{y})$ lies on the regression line y on x.

How did you do?

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4 marks

What to Watch (WTW) and Bingeable are two organisations that review and rank television series. Based on different sets of criteria, scores out of 5 are assigned to 6 recent television series (labelled A to F). The data is shown in the table below.

TV series	A	B	C	D	E	F
WTW’S score (x)	4.6	4.5	3.9	4.8	1.2	1.5
Bingeable’s score (y)	4.9	2.5	1.5	3.2	1.1	1.4
WTW’s rank
Bingeable’s rank

(i)

Find the Pearson’s product-moment correlation coefficient, r, for this data.

(ii)

Write down the equation of the regression line y on x.

How did you do?

2 marks

Fill in the two empty rank rows in the table above.

How did you do?

2 marks

Find the value of the Spearman’s rank correlation coefficient, r_s.

How did you do?

2 marks

Comment on the difference in the values of r and r_s.

How did you do?

Did this page help you?

4 marks

The table below shows the lengths, in km, of 5 taxi rides in Melbourne, Australia and the corresponding prices, in AUD.

Length, in km (x)	12.1	4.2	9.1	3.7	6.2
Price, in AUD (y)	26.75	5.75	8.50	5.50	6.95

Draw a scatter diagram for the above data on the axes below.

q1a-4-2-correlation-regression-veryhard-ib-ai-sl-maths-screenshot

How did you do?

3 marks

Calculate

(i)

$\bar{x}$ , the mean taxi ride length

(ii)

$\bar{y}$ , the mean price

(iii)

Plot the point $M (\bar{x}, \bar{y})$ on your scatter diagram.

How did you do?

3 marks

(i)

Write down the equation of the regression line y on x.

(ii)

Draw the regression line y on x on your scatter diagram.

How did you do?

1 mark

Show that the point $M (\bar{x}, \bar{y})$ lies on the regression line y on x.

How did you do?

Did this page help you?

4 marks

A health study was conducted on 5 male and 5 female participants, measuring their average daily caffeine intake, in milligrams (mg), and their resting heart rate, in beats per minute (BPM).

The following table shows the results of the study.

Average daily caffeine intake, in mg – male (x_m)	222	312	211	190	120
Resting heart rate, in BPM – male (y_m)	57	72	60	48	50
Average daily caffeine intake, in mg – female (x_f)	202	411	254	81	52
Resting heart rate, in BPM – female (y_f)	57	81	71	45	49

Calculate the Pearson’s product-moment correlation coefficient for,

(i)

the male participants, r_m,

(ii)

the female participants,r_f_.

How did you do?

4 marks

Write down the equation of the regression line

(i)

y_{m}

x_{m}

(ii)

y_{m}

x_{f}

How did you do?

3 marks

Find the intersection of the two regression lines found in part (b) and interpret its meaning.

How did you do?

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3 marks

The following table shows the distance, in km, to 5 different ferry destinations from Rostock, Germany and the corresponding price of the cruise, in €.

Destination	Copenhagen	Oslo	Stockholm	Helsinki	Riga
Distance, in km (D)	174	620	730	933	810
Price, in € (P)	30.50	65.00	45.75	85.50	125.00

The regression line P on D can be written in the form $P = a + b D .$

Calculate the values of a and b and interpret their meanings

How did you do?

2 marks

The distance to Aberdeen from Rostock is 1093 km.

Estimate the cost of the ferry to Aberdeen.

How did you do?

1 mark

Comment on the reliability of your estimate found in part (b).

How did you do?

3 marks

The ferry company decides to charge €135 to travel to Aberdeen.

Calculate the percentage error between your estimate found in part (b) and the actual price.

How did you do?

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2 marks

The following table shows the total revenue, R, in £, obtained weekly during the first 7 weeks of 2021 by Larry, an independent financial consultant, and the number of clients, x, served.

Week	1	2	3	4	5	6	7
Revenue, in £ (R)	2452	5751	6429	1203	4587	9786	6911
Clients, x	7	11	14	4	5	8	9

Write down the equation of the regression line R on x.

How did you do?

3 marks

Larry’s weekly operating costs are £2250 and the cost of serving each client is £35.

Find an expression for the profit Larry makes when serving x clients in a week.

How did you do?

3 marks

Estimate the least number of clients required to generate a minimum of £1000 profit.

How did you do?

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2 marks

Sandy Café is located on a beach and is open all year. The manager wants to see whether the daily average temperature, in °C, is correlated with the average tip they receive, as a percentage of the customer’s total bill. He records this data over 9 days and details it in the table below.

Daily average temperature, in °C (x)	22.4	27.8	15.4	12.2	8.8	2.1	33.4	14.7	19.4
Average tip as a percentage of the total bill (y)	20.1	16.3	12.4	12.8	10.1	9.4	18.8	13.1	15.9

(i)

Find the Pearson’s product-moment correlation coefficient, r, for this data.

(ii)

Write down the equation of the regression line y on x.

How did you do?

4 marks

On the 10^th day, the average temperature is 25° C and a customer tips their waiter $20.

Use the regression line to estimate the customer’s total bill. Give your answer to 2 decimal places.

How did you do?

4 marks

The customer’s total bill was $98.50.

(i)

Calculate the percentage error between your estimate found in part (b) and the actual total bill.

(ii)

Calculate the tip as a percentage of the actual total bill. Give your answer to the nearest whole number.

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1 mark

The table below shows the petrol prices, in New Zealand dollars (NZD) per litre, for 6 different petrol stations (labelled A to F) along with their distance south of Auckland’s city centre.

Petrol station	A	B	C	D	E	F
Distance south of Auckland, in km (x)	122	314	456	231	178	392
Petrol price, in NZD per litre (y)	1.94	1.88	1.78	1.84	1.99	1.81

Calculate the mean petrol price, $\bar{y}$ .

How did you do?

3 marks

The equation of the regression line y on x can be written in the form $y = a + b x$ .

(i)

Calculate the value of

a

(ii)

Calculate the value of b, giving your answer in the form

k \times 10^{n},

where

1 \leq | k | < 10, n \in ℤ

How did you do?

2 marks

The distance between Auckland’s city centre and a new petrol station, G, is 200 km and the bearing of G from Auckland’s city centre is 166°.

Estimate the petrol price at G.

How did you do?

3 marks

Petrol station G decides to set their price at 0.44 NZD per litre below the mean petrol price calculated in part (a).

Calculate the percentage error between the estimated petrol price for G calculated in part (c) and the actual petrol price at G.

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Correlation & Regression (DP IB Maths: AI SL)

Exam Questions

1. Number & Algebra

1.1 Number Toolkit

1.2 Sequences & Series

1.3 Financial Applications

2. Functions

2.1 Linear Functions & Graphs

2.2 Further Functions & Graphs

2.3 Modelling with Functions

3. Geometry & Trigonometry

3.1 Geometry Toolkit

3.2 Geometry of 3D Shapes

3.3 Trigonometry

3.4 Voronoi Diagrams

4. Statistics & Probability

4.1 Statistics Toolkit

4.2 Correlation & Regression

4.3 Probability

4.4 Probability Distributions

4.5 Binomial Distributions

4.6 Normal Distributions

4.7 Hypothesis Testing

5. Calculus

5.1 Differentiation

5.2 Integration

6. Extended Questions

6.1 Extended Questions