Exercises: Regression line

The equation of the regression line of weight ($Y$, in kg) on height ($X$, in cm) of a group of students is $y = 0,8x - 60$. Estimate the approximate weight of a student who is $170$ cm tall.

Calculate the equation of the regression line of $Y$ on $X$ for a bivariate distribution with the following known statistical parameters: the mean of $X$ is $\bar{x} = 5$, the mean of $Y$ is $\bar{y} = 12$, the variance of $X$ is $\sigma_x^2 = 4$ and the covariance is $\sigma_{xy} = 6$. Express the result in the form $y = mx + n$.

Observe the following scatter diagram showing the relationship between weekly study hours ($X$) and the mark obtained in an exam ($Y$). The blue line represents the regression line of $Y$ on $X$. Estimate graphically the approximate mark a student studying $7$ hours would obtain.

In a market study, the regression line of sales ($Y$) on advertising investment ($X$) is given by the equation $y = 2,5x - 4$. If it is known that the mean investment of the analysed companies is $\bar{x} = 6$, what is the value of the mean sales $\bar{y}$ of the distribution?

The linear correlation coefficient between the number of weekly training hours ($X$) and the time recorded in an obstacle race ($Y$) is $r = -0,92$. The regression line obtained is $y = -1,5x + 50$. The sample data correspond to athletes who train between $5$ and $15$ hours. Estimate the time for an athlete who trains $10$ hours and justify whether this prediction is considered reliable.

In a bivariate study, the equation of the regression line of $Y$ on $X$ is $y = 0,2x + 15$. Knowing that the covariance calculated for these data is $\sigma_{xy} = 3,5$, briefly justify whether the gradient of the line obtained is consistent with the sign of the covariance.

The relationship between two variables has been analysed, obtaining the equation $y = -3x + 20$ as the regression line of $Y$ on $X$. Is it possible for the linear correlation coefficient of this distribution to be $r = 0,85$? Give reasons for your answer based on the relationship between the gradient and the correlation.

The means of two statistical variables are $\bar{x} = 10$ and $\bar{y} = 25$. Check mathematically whether the line $y = 1,5x + 10$ can be a valid regression line for this data set, justifying your answer using a fundamental property that all regression lines must satisfy.

Based on data from AEMET in a mountainous region of the Iberian Peninsula, the relationship between the altitude of the municipality ($X$, in hundreds of metres) and its mean annual temperature ($Y$, in $^{\circ}\text{C}$) is analysed. Towns located between $400\text{ m}$ ($x = 4$) and $1800\text{ m}$ ($x = 18$) are recorded. The parameters obtained are: $\bar{x} = 10$ ($1000\text{ m}$), $\bar{y} = 12^{\circ}\text{C}$, $\sigma_x^2 = 16$, and the covariance is $\sigma_{xy} = -10,4$.

1. a) Before performing any algebraic calculations, explain what the sign of the slope should be and whether you expect the mean temperature in a town located at $1200\text{ m}$ ($x = 12$) to be higher or lower than the regional mean ($12^{\circ}\text{C}$).
2. b) Find the equation of the regression line of $Y$ on $X$.
3. c) Using the line, estimate the temperature of the town located at $1200\text{ m}$ and check whether your initial qualitative reasoning was correct.

Álvaro analyses a sample of cars from the Spanish market to study the relationship between engine power ($X$, in hp) and average fuel consumption ($Y$, in litres per $100\text{ km}$). He has the following summary data: $\bar{x} = 110\text{ hp}$, $\bar{y} = 5,8\text{ l/100 km}$, $\sigma_x^2 = 400$, and $\sigma_{xy} = 14$.

To obtain the regression line of $Y$ on $X$, Álvaro sets up:

From his result, he states: "For each additional horsepower of power, the car consumes about $28,6$ litres more per $100\text{ km}$".

Identify the conceptual error in Álvaro's formula, explain why his conclusion makes no practical sense whatsoever, and calculate the correct equation of the line, interpreting the new value of the slope properly.

A company records advertising expenditure $X$ (in thousands of euros) and sales revenue $Y$ (in thousands of euros) over $4$ consecutive months. The data pairs $(x_i, y_i)$ obtained are: $(2, 5)$, $(3, 7)$, $(4, 9)$ and $(5, 11)$. Calculate the equation of the regression line of $Y$ on $X$.

The regression line of the number of ice creams sold ($Y$) against the maximum daily temperature in $^\circ\text{C}$ ($X$) is $y = 15x - 120$. If a temperature of $30^\circ\text{C}$ is forecast for tomorrow, estimate how many ice creams will be sold. Furthermore, if on another day $255$ ice creams were sold, explain whether it is theoretically correct to use this same line to estimate the temperature on that day, or whether a rigorous procedure would require calculating a different line.

The following statistical parameters of a bivariate distribution are known: the correlation coefficient is $r = 0,8$, the standard deviation of $X$ is $\sigma_x = 2$, the standard deviation of $Y$ is $\sigma_y = 5$, and the centre of gravity is the point $(12, 40)$. From these data, calculate the equation of the regression line of $Y$ on $X$.

In a study on the length of an elastic spring ($Y$, in cm) when hanging different weights ($X$, in kg) from it, the two regression lines have been calculated. The regression line of $Y$ on $X$ is $y = 2,5x + 10$ and the regression line of $X$ on $Y$ is $x = 0,39y - 3,8$. Calculate the exact coordinates of the centre of gravity $(\bar{x}, \bar{y})$ of the distribution.

The relationship between the age of a car model ($X$, in years) and its second-hand market value ($Y$, in thousands of euros) has been studied. The regression line obtained is $y = -1,2x + 18$. If a car is $5$ years old, what is its estimated value? In addition, interpret the meaning of the y-intercept of the line in this real-life context.

Look at the two scatter diagrams, A and B, which show the relationship between two variables and include their respective regression lines. In which of the two distributions will making an estimate of $Y$ from a value of $X$ using the regression line be more reliable? Give reasons for your answer based on the visual concepts of correlation and dispersion.

In a class, the regression line of the Physics mark ($Y$) on the Mathematics mark ($X$) is $y = 0,9x + 0,5$. After reviewing the exams, the teacher detects a general error and decides to add $1$ extra mark to all Physics marks. What will be the new equation of the regression line of $Y$ on $X$? Give reasons for your answer.

Given the data table formed by the points $(1, 2)$, $(2, 4)$, $(3, 5)$ and $(4, 7)$, the equation of the regression line of $Y$ on $X$ has been calculated as $y = 1,6x + 0,5$. Calculate the residual (or estimation error) corresponding to the observation $x = 3$.

According to the Tourism Expenditure Survey (EGATUR) prepared by the INE, the relationship between the length of holiday stays ($X$, in days) and total expenditure per person ($Y$, in hundreds of euros) is analysed in a representative sample of international tourists in Spain, with stays between $3$ and $15$ days. The summary data are:

$\bar{x} = 8\text{ days}$, $\bar{y} = 11$ ($1100\text{ €}$),

$\sigma_x = 3\text{ days}$, $\sigma_y = 4$ ($400\text{ €}$), and $\sigma_{xy} = 10,8$.

1. a) Calculate the linear correlation coefficient $r$ and interpret whether the linear relationship between both variables is weak, moderate, or strong.
2. b) Determine the equation of the regression line of expenditure ($Y$) on days ($X$).
3. c) Estimate the total expenditure of a tourist whose planned stay is $11$ days and assess the reliability of this estimate.
4. d) A tour operator suggests using this same line to calculate the expenditure of a traveller who plans to stay for $60$ days. Using the concepts of interpolation and extrapolation, justify why it is not appropriate to apply the model in that case.

In a sociological study on youth leisure habits, an ink spill has erased the standard deviations and the covariance of two numerical variables, $X$ and $Y$. However, in the printed conclusions, the following data can be clearly read:

• The regression line of $Y$ on $X$ is $y = 1,5x + 4$.

• The mean of variable $X$ is $\bar{x} = 6$.

• The standard deviation of $X$ is $\sigma_x = 2$.

• The linear correlation coefficient is $r = 0,75$.

1. a) Deduce the coordinates of the centre of gravity of the distribution, $(\bar{x}, \bar{y})$.
2. b) From the slope of the line, find the value of the covariance $\sigma_{xy}$.
3. c) Reconstruct the value of the standard deviation of variable $Y$ ($\sigma_y$).
4. d) If $2$ fixed units were added to all observed values of $Y$ (keeping the values of $X$ unchanged), explain whether the slope of the regression line would increase, decrease, or remain constant.

The equations of the two regression lines of a bivariate distribution are known: $y = 0,5x + 2$ (regression line of $Y$ on $X$) and $x = 1,28y - 1,56$ (regression line of $X$ on $Y$). Calculate the linear correlation coefficient $r$, knowing that the correlation between both variables is positive.

In a bivariate distribution consisting of $N$ data values, the centre of gravity is the point $(4, 6)$ and the gradient of the regression line of $Y$ on $X$ is $1,5$. If a new data point that coincides exactly with the point $(4, 10)$ is added to the study, will the value of the gradient of the regression line change? Prove your answer mathematically by analysing how the formulas for covariance and variance change.

Prove analytically that if the linear correlation coefficient between two variables is perfect ($r = 1$ or $r = -1$), the equation of the regression line of $Y$ on $X$ and the equation of the regression line of $X$ on $Y$ coincide and represent the same geometric line.

The regression line of $Y$ on $X$ for an experiment has the equation $y = ax + b$. If, through a change of scale, we define a new variable $Z = 3X - 2$, determine the gradient of the new regression line of $Y$ on $Z$. Express the result in terms of the original parameter $a$.

In a laboratory test, the volume of a gas ($Y$) is measured at different temperatures ($X$). The regression line obtained after analysing the data is $y = 0,4x + 22$. Furthermore, it is known that the variance of the recorded temperatures is $\sigma_x^2 = 25$ and the variance of the volumes is $\sigma_y^2 = 5$. From this information, calculate the correlation coefficient $r$ and interpret whether the linear model is a good fit for these data.

We have a bivariate distribution with $100$ data values. The mean of $X$ is $50$ and the mean of $Y$ is $80$. The covariance is $120$ and the variance of $X$ is $60$. Following a quality review, it is discovered that one of the data pairs, specifically the point $(50, 80)$, was recorded incorrectly due to a sensor fault, and it is decided to remove it from the study. Calculate the new equation of the regression line of $Y$ on $X$ for the remaining $99$ data values and compare it with the original line.

Two statistical variables $X$ and $Y$ have been standardised, which means that their means are $\bar{x} = 0$ and $\bar{y} = 0$, and their standard deviations are $\sigma_x = 1$ and $\sigma_y = 1$. Express the equation of the regression line of $Y$ on $X$ using only the correlation coefficient $r$. What geometric shape does the line take if both variables are linearly independent?

In the analysis of a data set, it is observed that the regression line of $Y$ on $X$ is completely horizontal. What exact numerical value does the covariance $\sigma_{xy}$ have in this case? What statistical conclusion can we draw about the linear relationship between the variables $X$ and $Y$? Justify your answer using the general equation of the regression line.

A merchants' association in a Spanish city wants to forecast the annual revenue ($Y$, in thousands of euros) of new businesses based on the usable floor area of the premises ($X$, in $\text{m}^2$). The sample of registered establishments has areas between $50\text{ m}^2$ and $250\text{ m}^2$. To estimate the revenue of a new $140\text{ m}^2$ premises, three methods are debated:

• Method 1: Directly assign the value of the sample mean of revenue ($\bar{y}$) to any premises, regardless of its size.

• Method 2: Use the regression line of $Y$ on $X$ evaluated at $x = 140$, knowing that the sample correlation coefficient is $r = 0,82$.

• Method 3: Apply a proportional rule of three taking only the largest premises in the sample as a reference ($250\text{ m}^2$ and $180$ thousand euros in revenue).

1. a) Compare the advantages and limitations of each proposal and justify which one is the most solid and rigorous from a statistical point of view.
2. b) Explain what implicit geometric assumption Method 3 makes about the $y$-intercept and explain why it usually fails in economics.
3. c) If the sample correlation coefficient had been $r = 0,08$, explain whether you would still recommend the same method chosen in part a) or if one of the other two would become more reasonable.

A team of energy analysts crosses records from Red Eléctrica de España and AEMET during the summer to study daily electricity consumption for air conditioning ($Y$, in GWh) according to the day's maximum temperature ($X$, in $^{\circ}\text{C}$). Days with maximum temperatures between $24^{\circ}\text{C}$ and $42^{\circ}\text{C}$ are studied, obtaining:

$\bar{x} = 32^{\circ}\text{C}$, $\sigma_x = 4^{\circ}\text{C}$, $\bar{y} = 90\text{ GWh}$, $\sigma_y = 10\text{ GWh}$, and $\sigma_{xy} = 32$.

1. a) Obtain the regression line of $Y$ on $X$ and quantitatively interpret the practical meaning of its slope.
2. b) Calculate the linear correlation coefficient $r$ and assess the reliability of the model's predictions.
3. c) In the same report, a collaborator notices that the daily number of medical treatments for dehydration ($Z$) has a correlation of $r = 0,86$ with electricity consumption ($Y$), and proposes turning off air conditioning systems to reduce hospital emergency visits. Explain the logical fallacy of this proposal by clearly distinguishing between correlation and causation, and identify the confounding variable influencing both.

Topic challenge. Does screen use affect our rest? Investigate it by analysing data from your environment using a spreadsheet or GeoGebra.

1. Data collection and table: Ask $15$ people in your environment for their daily mobile phone hours ($X$) and sleep hours ($Y$). Prepare the two-dimensional frequency table and obtain the marginal distributions of both variables.

2. Conditional distributions: Calculate the conditional distribution of sleep hours for those who use screens more than $4$ hours daily and compare it with the overall mean. Are any differences visible at first glance?

3. Covariance and regression: In the digital tool, find the covariance ($\sigma_{xy}$) and interpret its sign. Obtain the regression line and provide a reasoned estimate of how much someone with $5,5$ hours of screen time would sleep.

4. Final report: Submit a short one-page report with the scatter plot, the fitted line, and your reasoned conclusions about the reliability of the prediction and the impact of mobile phones on rest.

Assessment criteria:

• Rigorous organisation of the two-way tables, marginal distributions, and conditional distributions.

• Correct mathematical calculation and interpretation of the covariance and the regression line using software.

• Consistency, clarity, and critical thinking in the report's conclusions based on the collected data.