Regression lines and reverse regression (x on y) + applications: notes and practice questions

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 $1.45\text{ tonnes}$.

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.

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 biologist is studying a species of fish. The length, $L$ cm, and weight, $W$ g, of each fish in a sample are recorded.

The lengths of the fish are summarized in the following box and whisker diagram.

Find the largest value of $L$ that would not be considered an outlier.

The regression line of $W$ on $L$ is $W = 25L - 150$. The regression line of $L$ on $W$ is $L = 0.03W + 10.5$.

One of the fish in the sample weighs 200 g. Estimate the length of this fish.

Find the mean weight of the fish in the sample.

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 study was conducted to investigate the relationship between the number of hours, $h$, a student spends studying for an exam and their score, $s$ (%), in that exam.

The number of hours spent studying is summarized in the following box and whisker diagram.

(a) Find the largest value of $h$ that would not be considered an outlier.

The regression line of $s$ on $h$ is $s = 5h + 25$. The regression line of $h$ on $s$ is $h = 0.1s + 2.5$.

(b) (i) One of the students scored 90% on the exam. Estimate the number of hours they studied.

(ii) Find the mean score of all the students in the study.

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 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.

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 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 botanist is investigating the effect of a new plant nutrient on the yield of a specific fruit crop. They apply varying amounts of the nutrient to several plants and record the amount of nutrient applied ($x$ grams per plant) and the resulting fruit yield ($y$ kg per plant). The results are shown in the table below.

The botanist wants to model the relationship between the amount of nutrient and the fruit yield using a linear regression. Find the equation of the regression line of $y$ on $x$.

Use your equation from part (a) to estimate the fruit yield for a plant that received $35$ grams of the nutrient.

Write down the correlation coefficient $r$.

Using the value of $r$, describe the correlation between the amount of nutrient and the fruit yield.

A marketing firm analyzes the relationship between advertising expenditure and monthly product sales. They find that the sales, $S$ (in thousands of units), are strongly correlated with the advertising expenditure, $A$ (in thousands of dollars). The line of best fit for $S$ on $A$ is of the form $S = mA + c$.

When the advertising expenditure is $A = 8$ thousand dollars, the estimated sales are $S = 120$ thousand units.

When the advertising expenditure is $A = 15$ thousand dollars, the estimated sales are $S = 232$ thousand units.

Find the values of:

i $m$

ii $c$.

State if there is positive or negative correlation.

The marketing firm plans an advertising expenditure of $A = 12$ thousand dollars. Find the estimated monthly sales, $S$.

When the advertising expenditure is $A = 10$ thousand dollars, find an estimate for the value of $S$, given that this is an interpolation.

A teacher wants to investigate the relationship between students' performance in Mathematics and Physics. They collect the final exam scores (out of 100) for 10 randomly selected students. The bivariate data obtained is given in the table below.

Student | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$ | $9$ | $10$

---|---|---|---|---|---|---|---|---|---|---

Math score ($x$) | $72$ | $80$ | $68$ | $75$ | $90$ | $83$ | $70$ | $95$ | $62$ | $88$

Physics score ($y$) | $75$ | $83$ | $70$ | $78$ | $92$ | $85$ | $73$ | $98$ | $65$ | $90$

(a) Find the Pearson product moment correlation coefficient, $r$, for this data.

(b) State, in two words, a description for this linear correlation.

(c) Find the equation of the line of best fit for:

i The Physics score ($y$) on the Math score ($x$).

ii The Math score ($x$) on the Physics score ($y$).

(d) Another student scored $86$ in Mathematics but was unable to take the Physics exam. Estimate the score they would have obtained in Physics, giving your answer to the nearest integer.

(e) A different student scored $77$ in Physics but did not take the Mathematics exam. Estimate the score they would have obtained in Mathematics, giving your answer to the nearest integer.

(f) If a student scored $110$ in Mathematics, explain why it would be unreliable to use a line of best fit to estimate their Physics score.

A researcher investigates the relationship between the number of hours spent studying per week, $h$, and the score on a standardized test, $s$, for a group of 40 students. The Pearson product-moment correlation coefficient is found to be $r = 0.75$, and the line of best fit for $s$ on $h$ is given by $s = 3.2h + 40$.

The researcher decides to adjust the data. The number of study hours for each student is reduced by 1, and the test score for each student is increased by 5.

(a) (i) State the new value of $r$ for the adjusted data.

(ii) State the new value for the gradient of the $s$ on $h$ line of best fit.

(iii) Give a reason for your answers to (i) and (ii).

(iv) Describe in two words the linear correlation that exists for this new data.

The original data, with $r = 0.75$ and $s = 3.2h + 40$, is now converted to different units. The study hours are converted from hours to minutes, and the test scores are scaled by a factor of 1.5.

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

(ii) Find the new value for the gradient of the line of best fit of the scaled scores on the scaled hours.

(iii) Give a reason for your answers to (i) and (ii).

Using the original data again ($r = 0.75$ and $s = 3.2h + 40$), a new variable called 'study deficit', $d$, is defined as $d = 20 - h$. The relationship between $s$ and $d$ is investigated.

(c) (i) State the new value of $r$ for $s$ and $d$.

(ii) Find the new line of best fit for $s$ on $d$.

(iii) Give a reason for your answers to (i) and (ii).

(iv) Describe in two words the linear correlation that exists for the new data.

A fitness researcher collected data on the average number of hours an individual spends exercising per week ($x$) and their average weekly calorie expenditure ($y$, in hundreds of calories). The results for a sample of individuals are shown in the table below.

This data can be modelled by the regression line with equation $y = ax + b$.

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

Explain what the gradient, $a$, represents in this context.

Use the model to estimate the average weekly calories burned if an individual exercises for $4.5$ hours per week.

Explain why it would be unreliable to use this model to predict the calories burned for someone exercising $15$ hours per week.

A research team is studying the relationship between the average daily temperature ($x$ in $^\circ C$) during a growing season and the average height of a specific plant species ($y$ in cm) at harvest. They collected data over seven growing seasons.

(a) Write down the equation of the $y$ on $x$ regression line for this data, giving your coefficients to three significant figures.

(b) Estimate the average plant height if the average daily temperature during the growing season was $25^\circ C$.

Give your answer to one decimal place.

(c) The average daily temperatures were converted from Celsius to Fahrenheit using the formula $F = \frac{9}{5}C + 32$.

For each of the following quantities, state whether it would change or remain the same:

(i) the mean of the average daily temperatures

(ii) the standard deviation of the average daily temperatures

(iii) the correlation coefficient, $r$

A local coffee shop records its weekly sales of two popular coffee beans, Espresso Blend and Single Origin, over 10 weeks. The sales figures, in kilograms (kg), are shown in the following table.

Weekly Sales (kg)

The cafe manager determines that the equation of the regression line of $y$ on $x$ for these sales is $y = 0.712x + 23.7$.

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

The new cafe manager uses the regression line of $y$ on $x$ ($y = 0.712x + 23.7$) for making predictions.

One week, Espresso Blend sales were $15$ kg. The manager estimated Single Origin sales to be $34$ kg. Give a reason why this estimation might not be appropriate.

Another week, Single Origin sales were $85$ kg. The manager used the line $y = 0.712x + 23.7$ to estimate Espresso Blend sales as $86$ kg. Give a reason why this method is not appropriate.

Use an appropriate method to show that the estimated Espresso Blend sales for the week when Single Origin sales were $85$ kg is $85$ kg (to the nearest integer).

A group of six students recorded the number of hours they spent studying for a mathematics exam, $x$, and their corresponding score on the exam, $y$, out of 100.

The data is presented in the table below:

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 $b$. Give your answers to three significant figures.

Use the equation of the regression line to estimate the exam score of a student who studies for $15$ hours per week.