Correlation & Regression (DP IB Analysis & Approaches (AA): 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 from Prague in the Czech Republic to eight different destinations in Europe.

Let $L_1$ be the regression line of $P$ on $D$. The equation of $L_1$ can be written in the form $P=aD+b$.

Let $L_2$ be the regression line of $D$ on $P$. The equation of $L_2$ can be written in the form $D=cP+d$.

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

(ii) Find the value of $c$ and the value of $d$.

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

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

The lines $L_1$ and $L_2$ both pass through the same point with coordinates $\left(p,q\right)$.

Find the value of $p$ and the value of $q$.

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.

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

The following table shows some of his findings.

Write down the independent variable and the dependent variable.

James believes that the relationship between $T$ and $t$ can be modelled by a linear regression equation. He describes the correlation as strong and positive.

Which value in the list below best represents the correlation coefficient?

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

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.

Easy Breezes is a company based in the snowy mountains of Greenland that makes heat pumps. The company wants to see whether the average weekly temperature, $x$ °C, is correlated with the average weekly energy consumption of its heat pumps, $y$ kWh. It records the following data.

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

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

(ii) Comment on the reliability of your estimate.

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.

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

(ii) Describe the correlation between the scores given by the two organisations.

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 $b$.

WTW gives a new series, G, a score of 4.7. Use the regression line of $y$ on $x$ to predict the score that Bingeable gives series G.

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

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

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

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

The distance from Rostock to Aberdeen is 1093 km.

Estimate the price of the ferry to Aberdeen.

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

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 $x$.

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 movie 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 $x$.

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.

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, 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.

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

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

Use the regression line from part (a) 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.

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

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 $1\le \left|k\right|<10,n\in \mathrm{\mathbb{Z}}$.

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.