Exercises: Covariance

In a two-dimensional study on $25$ individuals, the means $\bar{x} = 8$ and $\bar{y} = 15$ have been obtained. Knowing that the sum of the products of the variables is $\sum x_i y_i = 3200$, calculate the covariance $s_{xy}$.

Given the following table with three pairs of values corresponding to the variables $X$ and $Y$:

Calculate the center of gravity $(\bar{x}, \bar{y})$ and the covariance of the distribution.

In a sports performance investigation, the number of weekly training hours ($X$) is related to the time in minutes an athlete takes to complete a $10$ mile race ($Y$). What sign do you expect the covariance $s_{xy}$ to have? Briefly reason your answer.

For a set of $40$ two-dimensional data, it is known that $\bar{x} = 12$, $\bar{y} = 25$ and the covariance is $s_{xy} = -8,5$. Find the value of the sum of the products $\sum x_i y_i$.

In a thermodynamics experiment, the covariance between the pressure and the volume of a gas under certain conditions is $s_{xy} = -15,4$. If, to facilitate the calculations, the researcher decides to add $2$ to all the pressure values and subtract $5$ from all the volume values, what will be the new value of the covariance?

Calculate the covariance of the following two-dimensional distribution expressed in a table of absolute frequencies $f_i$:

When studying the relationship between the amount of fertilizer applied to a plant (in grams) and its weekly growth (in inches), a covariance of $24,6$ is obtained. What does the sign of this statistical parameter indicate about the relationship between the fertilizer and the growth?

The center of gravity of a two-dimensional distribution is $(5, 9)$. If the arithmetic mean of the products of the values of both variables is $\frac{\sum x_i y_i}{N} = 42$, determine the value of the covariance.

The temperature $X$ (in $^\circ\text{C}$) and the resistance $Y$ (in MPa) of a new plastic material have been measured in $5$ different tests, obtaining the following results:

Calculate the covariance of this two-dimensional distribution and interpret its sign in the context of the problem.

In an architecture study, the covariance between the length of a beam $X$ (measured in meters) and its maximum deformation $Y$ (measured in millimeters) is $s_{xy} = 3,5$. If it is decided to express the length in centimeters ($1\text{ m} = 100\text{ cm}$) and the deformation in micrometers ($1\text{ mm} = 1000\,\mu\text{m}$), calculate the new value of the covariance.

A survey has been carried out on $20$ families about the number of vehicles they own ($X$) and the number of parking spaces they own ($Y$). The results are summarized in the following two-way table:

Calculate the center of gravity and the covariance $s_{xy}$.

A set of $10$ pairs of two-dimensional data has means $\bar{x} = 6$ and $\bar{y} = 8$, and its covariance is $s_{xy} = 4$. If it is discovered that an eleventh pair of data is missing which coincides exactly with the center of gravity, that is, the pair $(6, 8)$, calculate the new covariance of the set of $11$ data.

Observe the following scatter graph representing four observations obtained in a laboratory experiment:

From the integer coordinates of the points shown in the graph, determine the table of values and calculate the covariance of the distribution.

In a study on screen time, the age of the respondents ($X$) has been grouped into intervals, relating it to the number of smart devices they own ($Y$):

Using the class marks for the variable $X$, calculate the covariance of the distribution.

From a two-dimensional statistical variable $(X,Y)$ composed of $8$ observations, the following data are known:

$\sum x_i = 40$, $\sum y_i = 64$, and $\sum x_i y_i = 280$.

Calculate the center of gravity and the covariance.

In an economic study on the relationship between the inflation rate ($X$, in $\%$) and the unemployment rate ($Y$, in $\%$) of a country over $5$ consecutive years, the following pairs of values were recorded: $(2, 8)$, $(3, 6)$, $(4, 5)$, $(5, 4)$, $(6, 2)$. Calculate the covariance and indicate whether the results support a direct or inverse relationship between both rates.

A new drug has been tested on a group of patients. The variable $X$ represents the administered dose (in mg) and the variable $Y$ the reduction in blood pressure (in mmHg). The results are shown in the following two-way table:

Calculate the covariance of this distribution and explain what the result suggests about the efficacy of the drug.

The trajectory of an object is analysed using $5$ points whose coordinates $(X,Y)$ are: $(-2, 4)$, $(-1, 1)$, $(0, 0)$, $(1, 1)$ and $(2, 4)$.

Calculate the covariance of this two-dimensional distribution. In view of the result, can it be stated that the variables $X$ and $Y$ are statistically independent? Justify your answer by relating it to the graph.

In a two-dimensional distribution of $3$ pairs of values, two of them are known: $(1, 2)$ and $(4, 6)$. The third pair is $(2, a)$, where $a$ is an unknown value. If it is known that the covariance of the distribution is $s_{xy} = 2$, set up an equation and find the value of $a$.

Given a two-dimensional variable $(X,Y)$, it is known that there is an exact linear functional relationship given by the equation $y_i = -3x_i + 5$ for all pairs in the distribution. Knowing that the variance of the variable $X$ is $s_x^2 = 4$, prove analytically and calculate the numerical value of the covariance $s_{xy}$.

In a company, the relationship between the employees' years of seniority ($X$) and their monthly salary in hundreds of dollars ($Y$) is studied. The following data are known: $\bar{x} = 10$, $\bar{y} = 20$, the variance of the seniority is $s_x^2 = 16$ and the covariance is $s_{xy} = 24$. It is decided to create a "total assessment" index defined as $Z = X + Y$. Calculate the covariance between the seniority ($X$) and this new index ($Z$).

A team of biologists has measured the weight $X$ (in kg) and the length $Y$ (in cm) of a reptile species, obtaining a covariance of $s_{xy} = 45$. For an international publication, they decide to transform the data: the weight will be expressed in pounds ($1\text{ lb} = 2,2\text{ lb}$) and the length in meters ($1\text{ ft} = 100\text{ in}$). Furthermore, due to an adjustment of the measuring equipment, they subtract $0,5\text{ ft}$ from all the lengths. What will be the value of the covariance with the new units?

A marketing department analyzes the advertising expenditure $X$ (in thousands of dollars) and the sales generated $Y$ (in thousands of dollars) in $10$ different campaigns. The following sums are available:

$\sum x_i = 150$, $\sum y_i = 800$, $\sum x_i^2 = 2500$, $\sum y_i^2 = 70000$, $\sum x_i y_i = 12500$.

Calculate the original covariance. If for the next year it is decided that all advertising expenditures will increase by $10\%$ and the expected sales will increase by $20\%$, what will be the estimated covariance of the transformed data?

A two-dimensional data set consists of the following $5$ strongly clustered points: $(1, 1)$, $(2, 2)$, $(2, 1)$, $(3, 2)$, $(3, 3)$; and an outlier: $(13, 15)$.

Calculate the covariance of the complete set of $6$ points. Next, calculate the covariance excluding the outlier and compare the results to explain the sensitivity of the covariance to extreme values.