Correlation & Regression (DP IB Applications & Interpretation (AI): HL): Exam Questions

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, $T$ °C, and the number of cups of tea sold, $C$. The results are shown in the following table.

The relationship between $T$ and $C$ can be modelled by the regression line of $C$ on $T$ with equation $C=aT+b$.

(i) Find the value of $a$ and the value of $b$.

(ii) Write down the value of Pearson's product-moment correlation coefficient, $r$.

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

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).

The following table shows the mean height, $y$ cm, of primary school children aged $x$ years.

The relationship between $x$ and $y$ can be modelled by the regression line of $y$ on $x$ with equation $y=ax+b$.

(i) Find the value of $a$ and the value of $b$.

(ii) Write down the value of Pearson's product-moment correlation coefficient, $r$.

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

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

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 number of calories, $y$, that she burnt during the run.

The number of calories burnt during a run depends on the length of the run. The relationship between $x$ and $y$ can be modelled by the regression line of $y$ on $x$ with equation $y=ax+b$.

(i) Find the value of $a$ and the value of $b$.

(ii) Write down the value of Pearson's product-moment correlation coefficient, $r$.

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

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

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

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

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

The relationship between $A$ and $V$ can be modelled by the regression line of $V$ on $A$ with equation $V=aA+b$.

(i) Find the value of $a$ and the value of $b$.

(ii) Write down the value of Pearson's product-moment correlation coefficient, $r$.

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

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

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

The price, $P$ US dollars, of an airline ticket depends 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.

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

(i) Write down the equation of the regression line of $P$ on $d$.

(ii) Write down the value of Pearson's product-moment correlation coefficient, $r$.

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

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

Compare this price to your answer to part (b), suggesting possible reasons for any difference.

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

The company would like to find Spearman's rank correlation coefficients for these volunteers' scores.

Complete the following table, ranking each volunteer's scores so that the lowest score has rank 1.

Find the value of 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.

Comment, in the context of the question, on each of your answers to part (b)(i), (ii) and (iii).

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

Explain, with a reason, whether this will change any of the values of Spearman's rank correlation coefficient, $r_s$, that you found in part (b).

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

The scatter diagram for these scores is shown below.

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

(i) with student J included

(ii) without student J included.

Complete the following table, ranking the scores in each subject so that the highest score has rank 1.

Find the value of Spearman's rank correlation coefficient, $r_s$,

(i) with student J included

(ii) without student J included.

Comment on your answers to parts (a) and (c).

A school decides to introduce a test for students aged 14, as they start their Mathematics course, to predict how well they will perform in their final Mathematics examination at age 16. Ten students are randomly selected to sit the test, and their results are compared with their final exam scores.

The results of the 10 students are shown in the table below.

State the name for this test of validity.

Calculate Pearson's product-moment correlation coefficient for this data.

Hence determine whether the test is a valid indicator of future performance in the final Mathematics examination. Give a reason for your answer.

A company is considering 8 candidates for a role and asks them to complete a set of questions designed to identify potential leadership qualities at the first stage of interview. A few days later, they ask them to complete a second set of similarly designed questions in order to test the reliability of the questionnaire.

The results have been recorded in the table below.

State the name of this test for reliability.

State a possible disadvantage of this test for reliability.

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

Hence determine whether the data from the questionnaire is reliable. Give a reason for your answer.

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

Find the range of the numbers of push-ups.

(i) Write down the value of Pearson's product-moment correlation coefficient, $r$.

(ii) Comment on the correlation between arm length and the number of push-ups.

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

Find the value of $a$ and the value of $b$.

Another student, Tom, can do 62 push-ups in one minute.

Explain why it would be inappropriate to use the regression line from part (c) to estimate Tom's arm length.

Give one other reason why an estimate of Tom's arm length from these data would not be reliable.

The following table shows the total ${\text{CO}}_2$ emissions, $T$ tonnes, of 5 countries (labelled A to E) and their average annual household income, $S$ US dollars.

(i) Find the value of Pearson's product-moment correlation coefficient, $r$.

(ii) Comment on the value of $r$.

The regression line of $S$ on $T$ can be written in the form $S=aT+b$.

Find the value of $a$ and the value of $b$.

State two reasons why it would be inappropriate to use the regression line from part (b) to estimate the total ${\text{CO}}_2$ emissions of a country where the average annual household income is 75 000 US dollars.

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.

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

Find the value of $a$ and the value of $b$.

Another student gave a cricket score of 6 but no football score.

Explain why it would be inappropriate to use the regression line from part (b) to estimate this student's football score.

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.

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

(i) Write down the value of Pearson's product-moment correlation coefficient, $r$.

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

(ii) Write down the value of $a$ and the value of $b$.

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

Explain why it would be inappropriate to use the regression line from part (b) to estimate the average number of hours of sleep of a participant whose score in the memory test is 67.

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.

Find

(i) $\overline{x}$, the mean length of the taxi rides;

(ii) $\overline{y}$, the mean price.

(iii) Plot the point $M\left(\overline{x},\overline{y}\right)$ on your scatter diagram.

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

(i) Find the value of $a$ and the value of $b$.

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

Show that the point $M\left(\overline{x},\overline{y}\right)$ lies on the regression line of $y$ on $x$.

Easy Breezes is a company in Greenland that installs heat pumps. The company wants to see whether the average weekly temperature, $x$ °C, is correlated with the average weekly energy used by one of its heat pumps, $y$ kilowatt hours (kWh). It records the following data.

(i) Calculate Pearson's product-moment correlation coefficient, $r$.

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

(i) Use your regression line from part (a)(ii) to estimate the energy used by one of the heat pumps in a week when the average weekly temperature is 11 °C.

(ii) Comment on the reliability of your estimate.

In a week when the average temperature was 11 °C, one of the heat pumps actually used 52.0 kWh.

Calculate the percentage error in your estimate from part (b)(i).

Sweet Dreams is a hotel ratings website. Its reviewers rate hotels on a scale of 1 to 10, and members of the public also leave ratings. The table below shows the ratings from Sweet Dreams and from the public for 9 different hotels (labelled A to I).

Complete the two empty rank rows in the table above, ranking the highest rating as 1.

(i) Calculate the value of Pearson's product-moment correlation coefficient, $r$.

(ii) Calculate the value of Spearman's rank correlation coefficient, $r_s$.

A mistake was made when recording the public's rating for Hotel D. The correct rating from the public for Hotel D is 2.3.

Explain why $r_s$ does not change when the public's rating for Hotel D is corrected.

The following table shows the distance, $D$ km, from Rostock, Germany, to 5 different ferry destinations, and the price, $P$ euros, of a ferry ticket to each one.

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

(i) Find the value of $a$ and the value of $b$.

(ii) Interpret the value of $b$ in the context of the question.

The distance from Rostock to Aberdeen is 1093 km.

Use your regression line to estimate the price of a ferry ticket to Aberdeen.

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

The ferry company charges 135 euros for a ticket to Aberdeen.

Calculate the percentage error in your estimate from part (b).

A supermarket has a physical store and an online store. The following table shows the number of customers, $x$, who came into the physical store, and the total revenue of the supermarket, $y$ US dollars, on each of 8 separate days.

(i) Find the value of Pearson's product-moment correlation coefficient, $r$.

(ii) Comment on the value of $r$.

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

Find the value of $a$ and the value of $b$, and interpret each value in context.

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 weight and their maximum sprint velocity. The data are recorded in the table below.

Find the value of Pearson's product-moment correlation coefficient, $r$,

(i) including players G and H;

(ii) excluding players G and H.

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

Find the value of $a$ and the value of $b$,

(i) including players G and H;

(ii) excluding 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.

Body mass index (BMI) is used to measure whether someone is overweight or underweight; however, BMI does not take a person's body fat percentage into account. The following table shows the BMI and the body fat percentage of 7 male participants.

(i) Find the value of Pearson's product-moment correlation coefficient, $r$.

(ii) Comment on the value of $r$.

The regression line of $y$ on $x$ can be written in the form $y=mx+c$.

Find the value of $m$ and the value of $c$, and interpret each value in context. Explain whether each interpretation is appropriate in the context of the question.

The formula to calculate a person's BMI is

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

John weighs 95 kilograms and is 1.84 metres tall.

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

A cinema is considering significantly reducing the price of its popcorn, as it believes that its customers spend more on drinks when they buy popcorn. It records the daily revenue from popcorn, $x$ US dollars, and the daily revenue from drinks, $y$ US dollars, on 8 randomly selected days.

Find

(i) $\overline{x}$, the mean daily revenue from popcorn;

(ii) $\overline{y}$, the mean daily revenue from drinks;

(iii) $r$, Pearson's product-moment correlation coefficient.

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

Find the value of $a$ and the value of $b$, and interpret each value in context. Explain whether each interpretation is appropriate in the context of the question.

Show that the point $M\left(\overline{x},\overline{y}\right)$ lies on the regression line of $y$ 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. 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.

Using your answer to part (b), comment on the reliability of Lara's exam, and give one reason why sitting the same exam the following day may not be a fair test of its reliability.

Lara thinks that the scores from the first exam can be used to predict the scores on 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 scores 60 on the first exam.

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.

The regression line of $R$ on $x$ can be written in the form $R=ax+b$.

Find the value of $a$ and the value of $b$.

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.

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 given to 6 recent television series (labelled A to F). The data is shown in the table below.

(i) Find Pearson's product-moment correlation coefficient, $r$, for this data.

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

Complete the following table, ranking each organisation's scores so that the highest score has rank 1.

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

Comment on the difference between the values of $r$ and $r_s$.

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.

Find the value of Pearson's product-moment correlation coefficient for

(i) the male participants, $r_m$;

(ii) the female participants, $r_f$.

The regression line of $y_m$ on $x_m$ can be written in the form $y_m=a_m{x}_m+b_m$, and the regression line of $y_f$ on $x_f$ in the form $y_f=a_f{x}_f+b_f$.

Find the value of

(i) $a_m$ and the value of $b_m$;

(ii) $a_f$ and the value of $b_f$.

Find the point of intersection of the two regression lines from part (b), and interpret it in context.

Sandy Café is located on a beach and is open all year. The manager wants to see whether the daily average temperature, $x$ °C, is correlated with the average tip customers leave, $y$, as a percentage of their total bill. He records the data over 9 days, as shown in the table below.

(i) Find Pearson's product-moment correlation coefficient, $r$, for this data.

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

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

Use the regression line to estimate the customer's total bill. Give your answer correct to two decimal places.

The customer's total bill was 98.50 US dollars.

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

The table below shows the petrol prices, in New Zealand dollars (NZD) per litre, at 6 different petrol stations (labelled A to F), and their distances south of Auckland's city centre.

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

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

(i) Calculate the value of $a$.

(ii) Calculate the value of $b$, giving your answer in the form $k\times {10}^n$, where $1\le \left|k\right|<10$ and $n\in \mathrm{\mathbb{Z}}$.

A new petrol station, G, is 200 km from Auckland's city centre, on a bearing of 166° from the city centre.

Estimate the petrol price at G.

Petrol station G decides to set its price at 0.05 NZD per litre below the mean petrol price found in part (a).

Calculate the percentage error between the estimated petrol price at G found 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. She uses specialist scientific equipment to heat an object to different precise temperatures, and uses the thermometer to read the temperature. To measure the reliability of the thermometer's readings, she takes two readings at each temperature, under the same conditions. The data is shown in the table below.

State the name of this test for reliability.

Determine whether the thermometer's readings are reliable by calculating 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 to measure temperature accurately.