Exercises: Regression line

The equation of the regression line of weight ($Y$, in pounds) on height ($X$, in inches) of a group of students is $y = 0.8x - 60$. Estimate the approximate weight of a student who is $170$ inches tall.

Calculate the equation of the regression line of $Y$ on $X$ for a two-dimensional distribution of which the following statistical parameters are known: 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 plot showing the relationship between weekly study hours ($X$) and the score obtained on an exam ($Y$). The blue line represents the regression line of $Y$ on $X$. Estimate graphically the approximate score a student who studies for $7$ hours would obtain.

In a market research 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 analyzed 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 course ($Y$) is $r = -0.92$. The regression line obtained is $y = -1.5x + 50$. The sample data corresponds to athletes who train between $5$ and $15$ hours. Estimate the time for an athlete who trains for $10$ hours and justify whether this prediction is considered reliable.

In a two-dimensional study, the equation of the regression line of $Y$ on $X$ is $y = 0.2x + 15$. Knowing that the covariance calculated for this data is $\sigma_{xy} = 3.5$, briefly justify whether the slope of the obtained line is consistent with the sign of the covariance.

The relationship between two variables has been analyzed and the equation $y = -3x + 20$ has been obtained as the regression line of $Y$ on $X$. Is it possible that the linear correlation coefficient of this distribution is $r = 0.85$? Justify your answer based on the relationship between the slope and the correlation.

The means of two statistical variables are $\bar{x} = 10$ and $\bar{y} = 25$. Determine mathematically whether the line $y = 1.5x + 10$ can be the valid regression line for this data set, justifying your answer using a fundamental property that all regression lines must satisfy.

A company records the advertising expenditure $X$ (in thousands of dollars) and the sales revenue $Y$ (in thousands of dollars) for $4$ 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 ($Y$) on 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 it was, or if the rigorous procedure would require calculating a different line.

The following statistical parameters of a two-dimensional 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 center of gravity is the point $(12, 40)$. From this data, calculate the equation of the regression line of $Y$ on $X$.

In a study on the length of an elastic spring ($Y$, in inches) when hanging different weights ($X$, in pounds) on 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 center of gravity $(\bar{x}, \bar{y})$ of the distribution.

The relationship between the age of a car model ($X$, in years) and its value in the used car market ($Y$, in thousands of dollars) has been studied. The regression line obtained is $y = -1.2x + 18$. If a car is $5$ years old, what is its estimated value? Furthermore, interpret what the y-intercept of the line means in this real-world context.

Observe the two scatter plots, A and B, which show the relationship between two variables and include their respective regression lines. In which of the two distributions will it be more reliable to make an estimate of $Y$ from a value of $X$ using the regression line? Justify your answer based on the visual concept of correlation and dispersion.

In a class, the regression line of the Physics grade ($Y$) on the Math grade ($X$) is $y = 0.9x + 0.5$. After reviewing the exams, the teacher detects a general error and decides to add $1$ extra point to all the Physics grades. What will be the new equation of the regression line of $Y$ on $X$? Justify your answer.

Given the data table formed by the points $(1, 2)$, $(2, 4)$, $(3, 5)$ and $(4, 7)$, it has been calculated that the equation of the regression line of $Y$ on $X$ is $y = 1.6x + 0.5$. Calculate the residual (or estimation error) corresponding to the observation $x = 3$.

The equations of the two regression lines of a two-dimensional 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 two-dimensional distribution formed by $N$ data points, the center of gravity is the point $(4, 6)$ and the slope 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 slope of the regression line change? Prove your answer mathematically by analyzing how the formulas for covariance and variance are modified.

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 $X$ of an experiment has the equation $y = ax + b$. If, due to a change of scale, we define a new variable $Z = 3X - 2$, determine what the slope of the new regression line of $Y$ on $Z$ will be. 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 analyzing 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 this data.

We have a two-dimensional distribution with $100$ data points. The mean of $X$ is $50$ and the mean of $Y$ is $80$. The covariance is $120$ and the variance of $X$ is $60$. After a quality review, it is discovered that one of the data pairs, specifically the point $(50, 80)$, was recorded incorrectly due to a sensor failure 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 points and compare it with the original line.

Two statistical variables $X$ and $Y$ have been standardized, 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.