Linear Correlation of bivariate data (scatter diagrams, lines of best fit, Pearson): notes and practice questions

An IB student is conducting five different investigations for their mathematics internal assessment. For each investigation, they collect bivariate data and calculate the Pearson product-moment correlation coefficient, $r$.

Use the list below to state the correct description for the following values of $r$.

perfect positive, strong positive, weak positive, zero, weak negative, strong negative, perfect negative

(a) For an investigation into daily temperature and the number of people at a public pool, $r = 0.91$.

(b) For an investigation into a car's engine size and its fuel efficiency, $r = -0.35$.

(c) For an investigation into the number of hours a person trains per week and their time to complete a marathon, $r = -0.82$.

(d) For an investigation into a person's shoe size and their monthly phone bill, $r = 0.03$.

(e) For an investigation into the side length of a square and its perimeter, $r = 1$.

An agronomist studies the effect of monthly rainfall on the yield of a new variety of wheat. The monthly rainfall, $R$ in mm, and the wheat yield, $Y$ in kg per hectare, were recorded for seven different regions.

The results are shown in the table below.

The relationship between the variables can be modelled by the regression equation $Y = aR + b$.

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

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

(c) Use the regression equation to estimate the wheat yield in a region where the monthly rainfall is 135 mm.

(a) A study investigates the relationship between the amount of a specific fertilizer ($x$ kg) applied to a crop field and the resulting crop yield ($y$ tonnes). Data from $8$ experimental plots is collected and presented in the table below.

(i) Calculate the Pearson product moment correlation coefficient for this data.

(ii) In two words, describe the linear correlation that is exhibited by this data.

(iii) Calculate the $y$ on $x$ line of best fit, in the form $y = ax + b$. Give the values of $a$ and $b$ to three significant figures.

(b) Another four experimental plots are added to the study, with the following results:

(i) Calculate the Pearson product moment correlation coefficient for the combined data of all $12$ plots.

(ii) In two words, describe the linear correlation that is exhibited by the combined data.

(iii) Suggest a reason why it would not be particularly valid to calculate the $y$ on $x$ line of best fit for the combined data.

The following table shows the advertising spending (in thousands of dollars) and the corresponding monthly sales (in thousands of units) for a new product over several months.

The data is also represented on the following scatter diagram.

The relationship between advertising spending (x) and monthly sales (y) can be modelled by the regression line of y on x with equation $y = ax + b$, where $a, b \in \mathbb{R}$.

Write down the value of $a$ and the value of $b$.

Use this model to predict the monthly sales (in thousands of units) when the advertising spending is 25 thousand dollars.

Write down the value of $\bar{x}$ and the value of $\bar{y}$.

Draw the line of best fit on the scatter diagram.

A marketing analyst is investigating the relationship between the amount spent on social media advertising, $x$ (in hundreds of dollars), and the monthly sales revenue, $y$ (in thousands of dollars), for a new product. The following data was collected over eight months:

The analyst models the data using the $y$ on $x$ regression line with equation $y = ax + b$.

Write down the values of $a$ and of $b$.

Explain what the gradient represents in the context of this problem.

Explain what the $y$-intercept represents in the context of this problem.

Estimate the monthly sales revenue if the company spends $350 on social media advertising.

The company aims to achieve a monthly sales revenue of $40000. Find how much they should spend on social media advertising, according to this model.

Explain why it might be inappropriate to use this model to predict the monthly sales revenue if the company spends $10000 on social media advertising.

Eight cars are tested to determine their fuel efficiency. Their weight, $W$, in tonnes, and their fuel efficiency, $F$, in kilometres per litre ($\text{km/L}$), are shown in the table.

The equation of the regression line of $F$ on $W$ for this data can be written in the form $F = aW + b$.

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

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

Use the equation of the regression line of $F$ on $W$ to predict the fuel efficiency of a car with a weight of $1.45\text{ tonnes}$.

A tutor wants to investigate the relationship between the number of hours a student spends studying for a mathematics test and the score they achieve on the test. They collect data from five students:

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

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

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

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

Another student studies for 6 hours for the mathematics test.

Use the regression line from part (a)(i) to estimate this student's test score.

A marine biologist is studying a species of sea turtle. She collects data on the carapace length, $L$ cm, and mass, $M$ kg, for 30 turtles. The Pearson's product-moment correlation coefficient for this data is found to be $r = -0.92$. The equation of the regression line of $M$ on $L$ is $M = -2.5L + 150$.

The biologist discovers her measuring tape was misaligned, and all length measurements are 2 cm too short. Her weighing scale was also faulty, showing a mass 1.5 kg less than the true mass for each turtle. The data is corrected for these errors.

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

(ii) State the new value for the gradient of the regression line of $M$ on $L$.

(iii) Briefly justify your answers to part (a)(i) and (a)(ii).

The biologist decides to present her findings to an international conference and converts her original measurements to different units. She converts the original length measurements from cm to mm, and the original mass measurements from kg to g.

(i) State the new value of $r$.

(ii) Find the new value for the gradient of the regression line of mass on length.

(iii) Briefly justify your answer for the new gradient.

For a different analysis, the biologist defines a "size index", $S$, as $S = 200 - L$. She investigates the relationship between the size index $S$ and the original mass $M$ in kg.

(i) Find the value of $r$ for the correlation between $S$ and $M$.

(ii) Find the gradient of the regression line of $M$ on $S$.

(iii) Describe the linear correlation between the size index $S$ and the mass $M$.

A coffee shop owner records the average daily temperature, $T$ (in °C), and the number of hot coffees sold, $C$, for a number of days. The scatter diagram shows the results.

The mean temperature for these days was 15 °C.

For these results, the equation of the regression line of $C$ on $T$ is $C = -5T + 250$.

(a) Find the mean number of hot coffees sold.

(b) Draw the regression line on the scatter diagram.

(c) By placing a tick (✔) in the correct box, determine which of the following statements is true.

(d) Give a reason why the regression line should not be used to estimate the number of hot coffees sold when the average temperature is 35 °C.

A botanist is studying the growth of a particular plant species. They record the average height of several plants (in cm) at different weeks after planting. The data collected is shown in the table below.

The relationship between the week number (x) and the plant height (y) can be modelled by the regression line of y on x with equation $y = ax + b$, where $a, b \in \mathbb{R}$.

Write down the value of $a$ and the value of $b$.

Use this model to predict the height of a plant after 15 weeks.

Write down the mean week number, $\bar{x}$, and the mean plant height, $\bar{y}$.

Draw the line of best fit on a scatter diagram for this data.

The following table shows the annual revenue of a tech startup, Quantum Innovations, $t$ years after its launch in 2015.

A data analyst uses linear regression to model the revenue of Quantum Innovations using these data.

The analyst's model is $R = at + b$.

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

(a)(ii) Interpret, in context, the value of $a$.

(b) The analyst uses this model to predict the revenue of Quantum Innovations in the year 2030, where $t = 15$, and calculates a revenue of approximately $11.8$ million USD.

Comment on the reliability of the analyst's prediction.

(c)(i) A financial expert, Elena, develops an exponential model for Quantum Innovations' future revenue.

In this model, $R_E(t) = 1.6(1.08)^t$ represents the revenue in millions of USD $t$ years after 2015, where $10 \le t \le 25$.

Use Elena's model to predict the revenue of Quantum Innovations in the year 2035.

(c)(ii) Interpret, in context, the value $1.08$ in Elena's model.

(d) Another financial expert, Carlos, develops a third model for Quantum Innovations' revenue.

In this model, $R_L(t) = \frac{20}{1+21e^{-0.15t}}$ represents the revenue in millions of USD $t$ years after 2015, where $10 \le t \le 25$.

Use Carlos's model to predict the revenue of Quantum Innovations in the year 2035.

(e) Determine the year in which the difference between the predictions from Elena's model and Carlos's model is greatest.

(f)(i) Find the value of

$R_E'(18)$;

(f)(ii) Find the value of

$R_L'(18)$.

(g) Compare and interpret, in context, the values of $R_E'(18)$ and $R_L'(18)$.

A university lecturer is investigating the relationship between the number of hours, $H$, students spend studying for a particular module each week and their final exam score, $S$, out of 120. The results for eight randomly selected students are summarized in the table below.

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

(b) The relationship between the variables can be modelled by the regression equation $S = aH + b$. Write down the value of $a$ and the value of $b$.

(c) One student, who currently studies 10 hours per week, decides to increase their study time by an extra three hours per week. Based on the given data, determine by how many marks their final exam score could be expected to change.

A tech company, "InnovateTech", is investigating the relationship between the average weekly training hours of its software developers and their quarterly productivity scores (out of 150). A sample of eight developers' data is collected and summarized in the table below.

Find Pearson's product-moment correlation coefficient, $r$, for these data.

The relationship between the variables can be modelled by the regression equation $P = ah + b$. Write down the value of $a$ and the value of $b$.

InnovateTech is considering providing an optional advanced training module. Based on the given data, determine how a developer's productivity score could be expected to alter if they completed this module, which adds an extra five hours of training per week.

The CEO of InnovateTech asserts that increased training hours directly cause higher productivity scores. Comment on the validity of the CEO's assertion.

InnovateTech later discovered that due to a data entry error, all recorded productivity scores were exactly 10 points lower than their true values. The data was corrected by adding 10 points to each developer's productivity score.

State how, if at all, the value of $r$ would be affected.

A rare vintage comic book, 'The Cosmic Crusader #1', was valued at $8000 on January 1st 2015. Its value is projected to increase by 3.5% on January 1st each year.

Find the projected value of 'The Cosmic Crusader #1' for the year 2025, to the nearest dollar.

Another rare comic book, 'Galactic Guardian #1', has had its value tracked over several years. The values for various years are shown in the following table.

Assuming 'Galactic Guardian #1''s annual value can be approximately modelled by the equation $V = ax + b$, use your GDC to show that 'Galactic Guardian #1' is projected to have a higher value than 'The Cosmic Crusader #1' in the year 2024, according to the model.

The total number of units produced, $P$, by a factory depends on the number of hours, $H$, the factory operates. A production manager uses the model $P = -0.8H^2 + 28H + 50$ to predict the total units produced on any given day, where $5 \le H \le 20$.

An energy auditor investigates the relationship between the total units produced and the energy consumption, $E$, in kilowatt-hours (kWh). The following table shows the data collected on five different days.

$\begin{array}{|c|c|} \hline \textbf{Units Produced (P)} & \textbf{Energy Consumption (E, in kWh)} \\ \hline 225 & 16.2 \\ 250 & 17.4 \\ 275 & 18.8 \\ 290 & 19.5 \\ 305 & 20.3 \\ \hline \end{array}$

Use the production model to estimate the number of units produced when the factory operates for 15 hours.

Find an appropriate regression equation that will allow the auditor to predict the energy consumption on a day when $P$ units are produced.

Hence, use your regression equation to predict the energy consumption when the factory operates for 15 hours.

A group of students recorded the number of hours they spent studying for a mathematics exam, $H$, and their corresponding exam score, $S$. The data is shown in the table below.

The regression line of $S$ on $H$ for this data can be written in the form $S = aH + b$.

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

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

Use your regression line to estimate the exam score of a student who studied for 9 hours.

A marketing analyst is investigating the relationship between the amount spent on online advertising and the number of product units sold.

The following table shows the advertising spend, $x$ (in thousands of dollars), and the corresponding number of units sold, $y$ (in hundreds), for a new product over seven different campaigns.

The value of Pearson's product-moment correlation coefficient, $r$, for this data is $0.999$, correct to three significant figures.

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

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

Use your regression line to estimate the number of units sold when the advertising spend is $12$ thousand dollars.

(a) A mathematics teacher investigates the relationship between the number of hours a student spends studying for a test and their final score on the test. The teacher collected data from five students, as shown in the table below:

The relationship between the hours studied, x, and the test score, 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.

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

(c) On another occasion, a student studied for 7 hours for the test.

Use the regression line from part (a)(i) to estimate this student's test score.

(a) The expected crop yield, $Y$, in kilograms (kg), from a certain field depends on the amount of fertilizer, $F$, applied in kg. A farmer models the relationship using the equation $Y = -0.8F^2 + 30F + 150$, where $5 \leq F \leq 25$.

Use this model to estimate the crop yield when the farmer applies 18 kg of fertilizer.

(b) The farmer also investigates the relationship between the crop yield, $Y$, and the total profit, $P$, in dollars ($). The following table shows the data collected from five different harvests.

Find an appropriate regression equation that will allow the farmer to predict the total profit based on the crop yield.

(c) Hence, use your regression equation to predict the total profit when the farmer applies 18 kg of fertilizer.