Correlation & Regression (DP IB Maths: AA HL)

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.

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

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

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.

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

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.

Use the regression line from part (b) to estimate the average number of hours of sleep a participant gets per night when their score in the memory test is 67. Give your answer to the nearest integer.

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

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

Country	A	B	C	D	E
emissions,  tonnes
Average annual household income, USD

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

Write down the equation of the regression line  on .

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.

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 , 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	-8.2	-4.3	-1.7	0.8	2.0	4.4	8.5
Average weekly energy consumption, in kWH	365.2	316.4	292.7	249.1	187.2	142.8	131.2

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

Write down the equation of the regression line  on .

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

Customers	45	88	54	97	154	107	36	72
Revenue, USD	548.21	832.55	497.71	1021.97	1138.73	988.62	1026.21	754.38

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

The regression line  on  can be written in the form .

Calculate the values of and and interpret their meanings.

The supermarket has daily fixed costs of 650 USD.

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

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

10 rugby players (labelled A to J) are used to investigate the relationship between a player’s maximum sprint velocity, in , 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	96	99	88	95	112	98	85	108	82	109
Maximum sprint velocity, in	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,

with players G and H.

Write down the equation of the regression line on 

with players G and H.

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.

A study was conducted on 6 students measuring their arm length,  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,  cm	72.2	69.2	75.6	78.1	78.5	74.5
No. of push ups,	34	42	24	30	38	31

State the range of the number of push ups.

Write down the equation of the regression line on .

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.

State whether your estimate is valid and justify your answer.

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

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

Comment on the result found for .

The regression line  on  is in the form .

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

The formula to calculate someone’s BMI is

John weighs 95 kilograms and is 1.84 metres tall.

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

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, $, and the daily revenue from drinks,  over 8 randomly selected days.

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

Calculate

, the mean daily revenue from popcorn.

, the mean daily revenue from drinks.

, the Pearson’s product-moment correlation coefficient.

The equation of the regression line  on  is in the form .

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

Show that the point lies on the regression line  on .

James decides to measure the performance of his laptop relative to its internal temperature. To measure his laptop’s performance James will record how long it takes, in seconds, to load a Safari browser window. To alter the temperature of his laptop James will start up some of the RAM intensive apps he has, which he knows causes the computer to heat up.

The following table details some of his findings

	39.2	42.8	50.1	55.6	68.7
(seconds)	2.2	3.2	3.1	4.1	5.2

Write down the independent and dependent variable.

James believes that the relationship between  and  can be modelled by a linear regression equation.

James describes the strength of the correlation as moderate. Circle the value below which best represents the correlation coefficient.

Suggest one significant drawback to James’ methods of testing the performance of his laptop relative to its internal temperature.

What to Watch (WTW) and Bingeable are two organisations that review 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	4.6	4.5	3.9	4.8	1.2	1.5
Bingeable’s score	4.9	2.5	1.5	3.2	1.1	1.4

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

Write down the equation of the regression line on .

WTW gives a new series G a score of 4.7. Use the regression line  on to predict the score that Bingeable awards the same series.

Comment on the reliability of your answer to part (c).

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

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

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

Calculate

, the mean taxi ride length

, the mean price

Plot the point on your scatter diagram.

Show that the point lies on the regression line on .

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

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

the male participants, ,

the female participants, .

Write down the equation of the regression line

on 

on .

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

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 ()	174	620	730	933	810
Price, in € ()	30.50	65.00	45.75	85.50	125.00

The regression line on can be written in the form .

Calculate the values of and and interpret their meanings

The distance to Aberdeen from Rostock is 1093 km.

Estimate the cost of the ferry to Aberdeen.

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

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

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

Write down the equation of the regression line  on .

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  clients in a week.

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

Sandy Café is located on a beach and is open all year. The manager wants to see whether the daily average temperature, in , 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	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	20.1	16.3	12.4	12.8	10.1	9.4	18.8	13.1	15.9

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

Write down the equation of the regression line  on .

On the 10^th day, the average temperature is 25  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.

The customer’s total bill was $98.50.

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

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	122	314	456	231	178	392
Petrol price, in NZD per litre	1.94	1.88	1.78	1.84	1.99	1.81

Calculate the mean petrol price, .

The equation of the regression line  on  can be written in the form .

Calculate the value of

Calculate the value of , giving your answer in the form , where .

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 .

Estimate the petrol price at G.