Correlation & Regression (DP IB Maths: AI 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.

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

(ii)     Write down the equation of the regression line  on .

(iii)    Draw the regression line on your scatter diagram.

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 A to E) and their average annual household income,  USD.

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

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

(ii)    Hence comment on the result.

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.

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).

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

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.

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

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

(ii)    Write down the equation of the regression line  on .

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

(ii)    Comment on the reliability of your estimate.

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.

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. 

The regression line  on  can be written in the form .

Calculate the values of a and b 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 x 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 ms^-1, and their weight, in kg. The data is recorded in the table below. 

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

Write down the equation of the regression line y on x

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, 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. 

State the range of the number of push ups.

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

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.

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

The regression line y on x is in the form .

Calculate the values of m and c 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, $x , and the daily revenue from drinks,$y over 8 randomly selected days.

Calculate 

, the mean daily revenue from popcorn. 

, the mean daily revenue from drinks. 

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

Calculate the values of a and b 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 y on x.

Lara, a biology teacher, wants to find out how much of the course her students have retained. She gives her students a practice exam and records their scores out of 100. In order to test the reliability of her exam she gives her students the same exam the following day. The results are shown below.

State the name of this test for reliability.

Find Pearson’s product moment correlation coefficient for the data in the table above.

Explain why Pearson’s product moment correlation coefficient might not be a good indication of the reliability of Lara’s exam.

Lara thinks that the scores from the first exam can be used to predict the results in the end of term exam which has different questions. The results are shown below.

Use a linear regression model to predict the end of term score for a student who gets a score of 60 on Lara’s exam.

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.

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

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

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

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

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

Calculate

Show that the point  lies on the regression line y on x.

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.

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

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 €.

The regression line P on D can be written in the form

Calculate the values of a and b 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 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.

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.

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

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.

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 °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.

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.

The customer’s total bill was $98.50.

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.

Calculate the mean petrol price, .

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

Calculate the value of 

Calculate the value of b, 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 166°.

Estimate the petrol price at G.

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.

Jezebel is testing the accuracy and consistency of her new thermometer when it measures the temperature of an object. To test her thermometer’s readings, Jezebel uses specialist scientific equipment to heat an object to different precise temperatures and uses the thermometer to read the temperature. She repeats this process twice using the same conditions to measure the reliability of her thermometer’s reading. The data is shown in the table below.

State the name of this test for reliability.

Determine whether the thermometer’s reading is reliable by calculating the Pearson’s product-moment correlation coefficient between relevant sets of data.

Explain why using only Pearson’s product-moment correlation coefficient is not necessarily a good measure of the criterion-related validity of the thermometer’s readings.

Explain whether or not Jezebel’s experiment is likely to have content validity for assessing the thermometer’s ability of measuring the temperature accurately.