Exercises: Regression line of y on x

Given the statistical parameters of a bivariate distribution: mean of $X$ equal to $4$, mean of $Y$ equal to $6$, variance of $X$ equal to $1.5$, and covariance equal to $3$. Determine the equation of the regression line of $Y$ on $X$.

In a study on the relationship between the amount of fertilizer applied ($X$, in kg/ha) and the yield obtained ($Y$, in metric tons/ha), the regression line of $Y$ on $X$ was calculated to be $y = 0.4x + 2.1$. Estimate the yield that will be obtained if $15$ kg/ha of fertilizer is applied.

Given the following table with three pairs of values of a bivariate distribution:

• $X$: $1 \quad 2 \quad 3$

• $Y$: $2 \quad 4 \quad 5$

Calculate the gradient of the regression line of $Y$ on $X$.

The regression line of $Y$ on $X$ of a bivariate distribution is $y = 1.2x - 0.5$. If we know that the marginal mean of the variable $X$ is $\bar{x} = 3$, what is the exact value of the marginal mean $\bar{y}$?

The following scatter plot shows the relationship between the years of experience of a group of technicians and their monthly salary in hundreds of pesos, together with the regression line of $Y$ on $X$. From the graph, visually estimate the expected salary for a technician with $6$ years of experience and determine the y-intercept of the line.

In a bivariate distribution, the standard deviations are known to be $\sigma_x = 2$ and $\sigma_y = 5$, and the linear correlation coefficient is $r = -0.8$. Calculate the equation of the regression line of $Y$ on $X$ knowing that the mean point of the distribution is $(10, 20)$.

The variables $X$ and $Y$ represent, respectively, the number of weekly training hours and the time in seconds to run $100$ meters for a group of athletes. If the regression line of $Y$ on $X$ is $y = -0.5x + 14$, interpret the physical meaning of the slope of this line in the context of the problem.

A bivariate distribution has as its data matrix the following summations: $\sum x_i = 20$, $\sum y_i = 40$, $\sum x_i^2 = 120$, $\sum x_i y_i = 180$, and the total number of observations is $N = 10$. Calculate the covariance and the variance of $X$, and write the equation of the regression line of $Y$ on $X$.

A digital marketing company has collected data on investment in social media campaigns ($X$, in thousands of pesos) and sales revenue generated ($Y$, in tens of thousands of pesos) over $5$ consecutive months:

• $X$: $50 \quad 60 \quad 70 \quad 80 \quad 90$

• $Y$: $240 \quad 280 \quad 300 \quad 360 \quad 380$

Calculate the regression line of revenue as a function of investment and estimate the expected revenue if $100\,000$ pesos is invested.

In a biomechanics study, the relationship between femur length ($X$, in cm) and height of an individual ($Y$, in cm) is analysed. The following statistical summaries are available for a sample of $50$ people:

$\bar{x} = 45$, $\bar{y} = 170$, $\sigma_x = 3.2$, $\sigma_y = 8.5$, $r = 0.92$.

Find the regression line of $Y$ on $X$ and estimate the height of a person whose femur measures $48$ cm. Briefly justify the reliability of this estimate.

The regression lines of a bivariate distribution are given by the equations $y = 0.5x + 4$ and $x = 1.6y - 2$. Determine the coordinates of the mean point of the distribution and the linear correlation coefficient $r$, giving reasons for its sign.

In a thermodynamics experiment, the temperature of a liquid ($Y$, in ºC) is measured as a function of elapsed time ($X$, in minutes). The regression line obtained is $y = -1.5x + 85$. Knowing that the coefficient of determination is $R^2 = 0.95$ and that the total duration of the experiment was $30$ minutes, is it reliable to estimate the temperature at $45$ minutes using this line? Justify your answer based on the concepts of interpolation and extrapolation.

The relationship between the amount of impurities in a metal alloy ($X$, in $\%$) and its tensile strength ($Y$, in MPa) has been studied. After analysing $20$ samples, $\bar{x} = 2.1$, $\bar{y} = 450$, and a regression line of $Y$ on $X$ given by $y = -15x + 481.5$ were obtained. Calculate the covariance of the distribution if the variance of the impurities is known to be $\sigma_x^2 = 0.8$.

The following scatter plot shows the relationship between the depth of a mineral seam ($X$, in meters) and the extraction cost per metric ton ($Y$, in thousands of pesos). The line plotted is the regression line of $Y$ on $X$. From the graph, obtain the coordinates of the two marked points on the line, calculate its exact equation, and estimate the extraction cost for a depth of $120$ meters.

The regression line of $Y$ on $X$ of a bivariate distribution is $y = 3x - 2$. If $4$ units are added to all observed values of the variable $X$ and all values of the variable $Y$ are multiplied by $2$, what will be the new equation of the regression line of $Y$ on $X$?

At a meteorological center, atmospheric pressure ($X$, in hPa) is related to the probability of precipitation ($Y$, in $\%$). Knowing that the mean point of the distribution is the point $(1013, 40)$, the standard deviation of the pressure is $5$ hPa, that of the probability is $20\%$, and the linear correlation coefficient is $-0.85$, find the equation of the regression line that allows the probability of rain to be estimated given the atmospheric pressure.

Prove analytically that if the variables $X$ and $Y$ have a perfect linear correlation ($r = 1$ or $r = -1$), then the regression line of $Y$ on $X$ coincides exactly with the regression line of $X$ on $Y$. Briefly explain what geometric implication this has for the angle formed by both lines in the plane.

In a study on the performance of solar panels, the variable $X$ represents solar irradiance ($W/m^2$) and the variable $Y$ the electric power generated ($W$). It is known that the regression line of $Y$ on $X$ is given by $y = 0.18x + 5$, and the regression line of $X$ on $Y$ by $x = 5y - 10$. Calculate the marginal means $\bar{x}$ and $\bar{y}$, and find the exact percentage of the variability of the electric power that is explained by the solar irradiance.

A set of bivariate data $(x_i, y_i)$ is available, where the variable $X$ takes the values $1, 2, 3, 4$ and the variable $Y$ takes the values $2, k, 5, 8$. Given that the gradient of the regression line of $Y$ on $X$ is $m_{yx} = 1.9$, set up an equation and determine the exact value of the parameter $k$.

In a macroeconomic analysis, the inflation rate ($X$, in $\%$) is related to the unemployment rate ($Y$, in $\%$). The regression line of $Y$ on $X$ is $y = -0.6x + 8$. If the marginal variance of inflation is $\sigma_x^2 = 4$ and the marginal variance of unemployment is $\sigma_y^2 = 2.25$, determine the linear correlation coefficient $r$ and the percentage of the variability of unemployment that is explained by the linear regression model.

A civil engineer is studying the relationship between the load supported by a beam ($X$, in metric tons) and its maximum deformation ($Y$, in millimeters). Some data from the original report has been lost, but it is known that the mean load is $\bar{x} = 15$, the variance of the load is $\sigma_x^2 = 16$, the covariance is $\sigma_{xy} = 18$, the correlation coefficient is $r = 0.9$ and the y-intercept of the regression line of $Y$ on $X$ is $b = 2$. Calculate the mean deformation $\bar{y}$, the standard deviation $\sigma_y$ and the equation of the regression line of $Y$ on $X$.

The following graph shows a scatter plot relating the speed of a vehicle ($X$, in km/h) to the braking distance ($Y$, in meters). The two regression lines have been drawn: Line A (in blue) and Line B (in green). Identify with reasoning, based on the properties of the slopes, which of the two lines is the regression line of $Y$ on $X$ and find its equation, knowing that both intersect at the point $(80, 40)$ and that the blue line passes through $(100, 56)$.

In a pharmaceutical study, the concentration of an active ingredient in blood ($Y$, in mg/L) is modelled as a function of the administered dose ($X$, in mg). If we change variables by standardising both variables, that is, we define $Z_x = \frac{X - \bar{x}}{\sigma_x}$ and $Z_y = \frac{Y - \bar{y}}{\sigma_y}$, prove algebraically that the regression line of $Z_y$ on $Z_x$ passes through the origin and that its gradient is exactly the linear correlation coefficient $r$.

A bivariate distribution of $100$ pairs of values has been studied. It is known that the sum of squared residuals (the sum of squared differences between the actual observed values $y_i$ and those estimated by the line $\hat{y}_i$) is equal to $150$. If the marginal variance of the variable $Y$ is $\sigma_y^2 = 4$, find the coefficient of determination $R^2$ using the relationship $\sum (y_i - \hat{y}_i)^2 = N \times \sigma_y^2 (1 - r^2)$, and calculate the absolute value of the gradient of the regression line of $Y$ on $X$ knowing that the marginal variance of $X$ is $\sigma_x^2 = 9$.