Two-Variable Statistics Problems

The regression line of $Y$ on $X$ of a bivariate distribution is given by the equation $y = 3.2x - 1.5$. If the arithmetic mean of the variable $X$ is $\bar{x} = 4.5$, determine the value of the arithmetic mean of the variable $Y$, $\bar{y}$.

In a study on the weight ($X$, in kg) and height ($Y$, in cm) of a group of students, a linear correlation coefficient $r = 0.85$ was obtained. If the marginal standard deviations are $S_x = 12$ and $S_y = 8$, calculate the exact value of the covariance $S_{xy}$.

The following statistical parameters are known for a bivariate distribution: $\bar{x} = 12$, $\bar{y} = 25$, the marginal variance of $X$, $S_x^2 = 16$, and the covariance $S_{xy} = 24$. Write down the equation of the regression line of $Y$ on $X$.

When studying the statistical relationship between two variables $X$ and $Y$, the following regression lines were obtained:

Line of $Y$ on $X$: $y = 0.5x + 4$

Line of $X$ on $Y$: $x = 1.5y - 3$

Calculate the coordinates of the mean point (center of gravity) of the bivariate distribution $(\bar{x}, \bar{y})$.

The regression lines of a bivariate distribution are given by the equations $y = -0.5x + 8$ (line of $Y$ on $X$) and $x = -1.28y + 10$ (line of $X$ on $Y$).

Calculate the linear correlation coefficient $r$ and find the marginal means $\bar{x}$ and $\bar{y}$.

The following scatter plot shows the relationship between the age of a group of vehicles ($X$, in years) and their current market value ($Y$, in thousands of pesos). By observing the shape of the scatter plot, state the sign that the covariance will have and whether the linear correlation between both variables is strong or weak.

To estimate monthly electricity consumption ($Y$, in kWh) based on the number of people living in a household ($X$), the regression line $y = 45x + 120$ was obtained with a correlation coefficient $r = 0.92$. Estimate the consumption for a household with $4$ people and state whether the estimate is very reliable or unreliable, briefly justifying your answer.

In a study relating the marketing budget of several companies ($X$) to their annual profits ($Y$), the linear correlation coefficient has been calculated as $r = 0.7$. What percentage of the variability in profits is explained by the marketing budget according to the proposed linear regression model?

The accompanying scatter diagram represents the relationship between the monthly mean temperature ($X$, in $^\circ\text{C}$) and the consumption of natural gas ($Y$, in $\text{m}^3$) in a city during the last year. Given that the marginal means are $\bar{x} = 15$ and $\bar{y} = 120$, and that the gradient of the regression line of $Y$ on $X$ is $-3.5$, find the equation of this line and estimate the gas consumption for a month with a mean temperature of $8 \ ^\circ\text{C}$.

A study has been carried out on the number of weekly training hours ($X$) and the time achieved in an athletics event ($Y$, in seconds) for a group of runners. Following the analysis, it is known that $\bar{x} = 12$, $\bar{y} = 45$, $S_x = 3$, $S_y = 2$ and the covariance is $S_{xy} = -4.8$.

Calculate the regression line of $X$ on $Y$ and estimate how many weekly hours an athlete trained who achieved a time of $42$ seconds in the event.

The time in hours ($X$) and the distance covered in km ($Y$) by a cyclist over $5$ stages have been measured, obtaining the following pairs of values $(x_i, y_i)$: $(2, 45)$, $(3, 65)$, $(4, 85)$, $(2, 40)$ and $(5, 110)$. Calculate the covariance of this bivariate distribution.

In a physics laboratory, the extension of an elastic spring ($Y$, in cm) was measured when hanging different masses ($X$, in kg) from it. The results obtained in the experiment are:

$X$: $1$, $2$, $3$, $4$, $5$, $6$

$Y$: $2.1$, $3.9$, $6.2$, $7.8$, $10.1$, $12.2$

Calculate the linear correlation coefficient $r$ and the equation of the regression line of $Y$ on $X$.

A technology company wants to predict the number of sales of a new smartphone ($Y$, in thousands of units) based on daily social media mentions ($X$, in thousands). Data from $8$ previous launches have been collected, obtaining the following summations:

$\sum x_i = 160$, $\sum y_i = 400$, $\sum x_i^2 = 3600$, $\sum y_i^2 = 21500$, $\sum x_i y_i = 8600$.

Calculate the covariance, the linear correlation coefficient, and estimate the sales that will be obtained if $25$ thousand mentions are recorded on the launch day.

For a bivariate distribution, the following data are known: sample size $n = 10$, marginal means $\bar{x} = 5$ and $\bar{y} = 8$, marginal variances $S_x^2 = 4$ and $S_y^2 = 9$. Furthermore, the equation of the regression line of $Y$ on $X$ is $y = 1.2x + 2$.

Calculate the exact value of the covariance $S_{xy}$ and the linear correlation coefficient $r$.

A factory is studying the relationship between the age of its machines ($X$, in years) and the annual maintenance cost ($Y$, in hundreds of pesos). Data has been collected from $4$ machines, but due to a computer error, the maintenance cost of one of them has been lost. The pairs $(x_i, y_i)$ are: $(2, 80)$, $(3, 100)$, $(5, k)$, $(6, 180)$.

Knowing that the marginal mean of the maintenance cost for the four machines is $\bar{y} = 130$, determine the value of $k$ and then calculate the equation of the regression line of $Y$ on $X$.

The following contingency table shows the data for $100$ ceramic items, classified by size ($X$: Small=$1$, Large=$2$) and quality ($Y$: Defective=$0$, Optimal=$1$). Write down the table for the marginal distribution of the variable $Y$ and calculate its mean $\bar{y}$.

The following two-way table shows the distribution of $50$ families according to the number of children ($X$) and the number of vehicles they own ($Y$):

Calculate the mean and standard deviation of the conditional distribution of the number of vehicles owned by families, knowing that they have exactly $1$ child (that is, the distribution $Y|X=1$).

The following joint frequency table (absolute frequencies) is given for two qualitative variables $X$ and $Y$:

Determine the exact value of the parameter $k$, knowing that the variables $X$ and $Y$ are statistically independent.

The correlation coefficient between monthly income ($X$, in pesos) and expenditure on leisure ($Y$, in pesos) for a group of people is $r = 0.75$. If we decide to express income in thousands of pesos ($X' = \frac{X}{1000}$) and increase leisure expenditure by a fixed amount of $1\,000$ pesos due to a subsidy ($Y' = Y + 1\,000$), what will be the new linear correlation coefficient between $X'$ and $Y'$? Justify your answer using the properties of covariance and standard deviations.

In a statistical study on two variables $X$ and $Y$, a change of variables was performed by standardising or rescaling the original data in the form $U = \frac{X - 10}{5}$ and $V = \frac{Y - 20}{2}$. It is known that the regression line of $V$ on $U$ is $v = 0.8u + 1.5$.

By reversing the change of variables, find the equation of the regression line of $Y$ on $X$ in explicit form $y = mx + n$.

The following scatter diagram shows the growth of a bacterial colony ($Y$, in thousands) as a function of time ($X$, in hours). The scatter plot suggests an exponential growth of the type $Y = A \times e^{BX}$.

To linearise the model, it has been proposed to make the change of variable $Z = \ln(Y)$. If, by performing a linear regression fit of $Z$ on $X$, we obtain the line $z = 0.4x + 1.2$, determine the numerical values of the constants $A$ and $B$ of the original exponential model.

Prove mathematically, using the expressions for the gradients of the regression lines as a function of the covariance and the marginal variances, that if the linear correlation coefficient is perfect ($r = 1$ or $r = -1$), the two regression lines (that of $Y$ on $X$ and that of $X$ on $Y$) are coincident lines.

In a statistical study, the regression line of $Y$ on $X$ is $y = 2x - 3$ and the regression line of $X$ on $Y$ is $x = 0.4y + 2$. Knowing that the marginal variance of the variable $X$ is $S_x^2 = 5$, calculate the value of the marginal variance of $Y$, $S_y^2$, and the value of the covariance $S_{xy}$.

Two groups of employees from the same multinational company have been studied to analyse the relationship between years of experience ($X$) and monthly salary ($Y$).

In Group A, consisting of $10$ employees, the following were obtained: $\bar{x}_A = 5$, $\bar{y}_A = 1500$, and a covariance $S_{xy,A} = 200$.

In Group B, consisting of $20$ employees, the following were obtained: $\bar{x}_B = 5$, $\bar{y}_B = 1500$, and a covariance $S_{xy,B} = 350$.

Calculate the total covariance $S_{xy}$ if we decided to group all $30$ employees into a single bivariate distribution.