Exercises: Bivariate distributions

Classify the following relationships between bivariate variables as deterministic (functional) dependency or statistical (random) dependency:

1. a) The radius length of a circle and its area.
2. b) The number of training hours of an athlete and their time in a $100$-metre race.
3. c) The weight of a bag of oranges and its price in the supermarket.
4. d) The age of a tree and its height.

In a study on climate and habits across several cities, different pairs of variables have been analysed. Indicate whether a positive, negative, or zero correlation is expected in each case:

1. a) The altitude of the city and the mean annual temperature.
2. b) The annual rainfall amount and the number of umbrellas sold.
3. c) The population and the mean annual temperature.
4. d) The annual sunshine hours and heating expenditure.

For a bivariate distribution $(X, Y)$ of $10$ pairs of values, the following sums have been obtained after tabulating the data: $\sum x_i = 40$, $\sum y_i = 60$ and $\sum x_i y_i = 280$. Calculate the covariance of the distribution.

The regression line relating weekly study hours ($X$) to the mark obtained on an exam out of $10$ ($Y$) is given by the equation $y = 0,6x + 2,5$.

1. a) What mark is estimated for a student who studied $8$ hours?
2. b) If a student wants to obtain an estimated mark of $7$, how many hours should they study according to this model?

Match each of the following values of Pearson's linear correlation coefficient ($r$) with the corresponding graph in the diagram below.

Values: $r_1 = 0,98$, $r_2 = -0,92$, $r_3 = 0,05$, $r_4 = 0,45$.

The weight in kg ($X$) and height in cm ($Y$) of a group of $5$ friends have been measured, obtaining the following pairs of values: $(60, 165)$, $(65, 170)$, $(70, 175)$, $(55, 160)$ and $(80, 180)$. Calculate the exact coordinates of the centre of gravity $(\bar{x}, \bar{y})$ of this bivariate distribution.

In a laboratory experiment, we want to test how temperature ($T$) affects the solubility of a salt in water ($S$).

1. a) Identify which variable should be considered the independent variable ($X$) and which the dependent variable ($Y$) to conduct a regression study.
2. b) If as temperature increases, the amount of dissolved salt increases proportionally, what sign will the linear correlation coefficient have?

The linear correlation coefficient between shoe size for a group of students and their mark on the latest mathematics exam is $r = 0,02$. Explain the meaning of this value. Does it make sense to calculate the regression line to predict the mark from shoe size?

In a road safety project, Lucía studies two possible factors influencing a vehicle's braking distance ($Y$, in meters): speed ($X_1$, in $\text{km/h}$) and the driver's reaction time ($X_2$, in seconds).

After analyzing the sample data, she obtains a covariance $\sigma_{x_1 y} = 320\text{ km/h}\cdot\text{m}$ for the first factor and $\sigma_{x_2 y} = 12\text{ s}\cdot\text{m}$ for the second. From this, Lucía states:

"The linear relationship between speed and braking is almost 27 times stronger than between reaction time and braking because $320$ is much greater than $12$."

In addition, she adds: "If we convert speed to meters per second by dividing by $3,6$, the correlation coefficient $r$ will also be divided by $3,6$."

1. a) Identify and explain, with reasoning, the two conceptual errors that Lucía made in her statements.
2. b) Knowing that in the speed study the standard deviations are $\sigma_{x_1} = 20\text{ km/h}$ and $\sigma_y = 18\text{ m}$, calculate the true linear correlation coefficient $r$ and interpret the strength and direction of the relationship. What value would $r$ take if speed were measured in $\text{m/s}$?

The number of weekly study hours ($X$) and the grade obtained in an exam out of 10 ($Y$) have been recorded for five $4.º\text{ de ESO}$ students:

• Student A: $(2; 3)$

• Student B: $(4; 5)$

• Student C: $(6; 6)$

• Student D: $(8; 7)$

• Student E: $(10; 9)$

1. a) Make a reasoned estimation before calculating:

1. Looking at the extreme data points and their symmetry, propose an estimated value for the coordinates of the centroid (mean point) $(\bar{x}, \bar{y})$.

2. Observe whether both variables increase together and reasonably estimate the sign of $r$ and whether its value will be closer to $0$, $0,5$, or $1$.

1. b) Check your estimates by calculating the exact means $\bar{x}$ and $\bar{y}$, the variances $\sigma_x^2$ and $\sigma_y^2$, the covariance $\sigma_{xy}$, and the correlation coefficient $r$ (to two decimal places). Finally, assess whether your initial estimates were accurate.

In a market study on house prices ($Y$, in thousands of euros) based on floor area ($X$, in square metres), the following statistical parameters were obtained:

• The covariance is $\sigma_{xy} = 450$.

• The standard deviation of floor area is $\sigma_x = 25$.

• The standard deviation of price is $\sigma_y = 20$.

Calculate the linear correlation coefficient and interpret the strength and direction of the relationship between both variables.

A total of $6$ families were surveyed about their monthly income in thousands of euros ($X$) and their monthly expenditure on leisure in hundreds of euros ($Y$). The results are the pairs:

$(2, 3)$, $(2,5; 4)$, $(3, 4)$, $(4, 6)$, $(4,5; 7)$, $(5, 8)$.

1. a) Plot the scatter diagram on coordinate axes.
2. b) By observing your plot, estimate whether the correlation is strong or weak, and whether it is positive or negative.
3. c) Without doing complex regression calculations, do you think the regression line would pass close to the point $(3,5; 5,3)$? Justify your answer by calculating the centre of gravity of the distribution.

We know the following data for a bivariate distribution concerning a company's sales ($Y$, in thousands of units) based on advertising investment ($X$, in thousands of euros):

• Centre of gravity: $(12, 40)$.

• Standard deviation of $X$: $\sigma_x = 3$.

• Standard deviation of $Y$: $\sigma_y = 5$.

• Correlation coefficient: $r = 0,9$.

Find the equation of the regression line of $Y$ on $X$ in the form $y = mx + n$.

The regression line relating the age of a car model in years ($X$) to its second-hand market value in thousands of euros ($Y$) is $y = -1,2x + 18$.

1. a) Estimate the value of a car that is $5$ years old.
2. b) Estimate the value of a car that is $20$ years old.
3. c) Are both estimates equally reliable? Explain the logical and mathematical risks of extrapolation in the second case.

A statistical study conducted over a year in a coastal town shows a strong positive correlation between monthly ice cream sales and the number of people involved in drowning incidents at the beach.

1. a) Does this mean that eating ice cream causes drownings?
2. b) Identify a possible "hidden variable" that explains the simultaneous increase in both variables and justifies the observed statistical correlation.

A small software company recorded the number of developers assigned to a project ($X$) and the time in months taken to complete it ($Y$):

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

$Y$: $8, 6, 3, 2$

1. a) Calculate the mean of $X$ and the mean of $Y$.
2. b) Calculate the covariance of the distribution.
3. c) Is the relationship direct or inverse? Does this result make sense in the context of the problem?

In a 4th ESO class, the relationship between Physics marks ($X$) and Mathematics marks ($Y$) of the $20$ enrolled students was studied. The centre of gravity of the distribution is $(6,2; 6,5)$.

If a new student joins the class and achieves a mark of $4$ in Physics and $5$ in Mathematics, how will the centre of gravity of the new distribution of $21$ students shift? (You do not need to calculate the exact value, but reason the direction of change in both coordinates).

Given the equations of the regression line of $Y$ on $X$ ($y = 0,8x + 2,4$) and the regression line of $X$ on $Y$ ($x = 0,5y + 3$) for the same bivariate distribution, determine the coordinates of the centre of gravity $(\bar{x}, \bar{y})$. (Recall the geometric property relating both lines to the centre of gravity).

In a sociological report, the relationship between monthly net income ($X$, in thousands of euros) and monthly expenditure on leisure ($Y$, in hundreds of euros) was studied for a group of families. Due to a computer error, the table with the individual data was lost, but the equations of the two regression lines were preserved in the conclusions:

• Regression line of $Y$ on $X$: $y = 0,75x + 2$

• Regression line of $X$ on $Y$: $x = 0,48y + 1,6$

1. a) Find, with justification, the coordinates of the centroid (mean point) $(\bar{x}, \bar{y})$ of the distribution without needing the original data.
2. b) Determine the linear correlation coefficient $r$ from the slopes of both lines. Justify the sign assigned to $r$.
3. c) Knowing that the variance of income is $\sigma_x^2 = 16$, reconstruct the exact value of the covariance $\sigma_{xy}$ and that of the standard deviation of leisure expenditure $\sigma_y$.

A geography team uses AEMET records on a clear winter day to model how the maximum temperature ($Y$, in ºC) decreases as altitude ($X$, expressed in hundreds of metres) increases across five locations in the Sierra de Guadarrama:

• Collado Villalba: $(9; 11)$

• Cercedilla: $(12; 9)$

• Navacerrada town: $(13; 6)$

• Puerto de Cotos: $(18; 2)$

• Puerto de Navacerrada: $(23; -3)$

(For example, $X = 9$ represents an altitude of $900\text{ m}$).

1. a) Model the linear relationship by calculating the covariance $\sigma_{xy}$ and the linear correlation coefficient $r$ (to three decimal places). What does this coefficient indicate about the goodness of fit?
2. b) Determine the equation of the regression line of $Y$ on $X$ and interpret the physical meaning of its slope (the vertical temperature lapse rate) in the mountain context.
3. c) Use the line to estimate the maximum temperature at Mirador de las Canchas ($1750\text{ m}$, that is, $X = 17,5$) and at the summit of Peñalara ($2428\text{ m}$, that is, $X = 24,28$). Reason which of the two estimates is more reliable considering the observed range of data.

A transport company wants to analyse the relationship between the distance covered on a journey ($X$, in tens of km) and fuel consumption ($Y$, in litres). The data recorded for a sample of $5$ lorries are:

$X$: $5, 10, 15, 20, 25$

$Y$: $12, 22, 30, 42, 49$

1. a) Calculate the means $\bar{x}$ and $\bar{y}$.
2. b) Calculate the variances $\sigma_x^2$ and $\sigma_y^2$, and the covariance $\sigma_{xy}$.
3. c) Find Pearson's linear correlation coefficient $r$.
4. d) Determine the equation of the regression line of $Y$ on $X$ and estimate fuel consumption for a journey of $180$ km ($X=18$).

The relationship between weekly training hours ($X$) and the time in minutes taken to complete an endurance sporting event ($Y$) in a group of athletes has been studied, yielding the following scatter diagram.

When calculating the correlation coefficient with all the data, a weak correlation is obtained ($r = -0,45$).

1. a) Observing the graph, identify which point is considered an outlier.
2. b) Reasoning about the concept of linear trend, explain how removing "Athlete A" would affect the numerical value of the correlation coefficient and the slope of the regression line.

In a small town, information was collected on the number of cars per family ($X$) and the number of televisions per family ($Y$), presented in the following two-way table:

1. a) Obtain the marginal distributions of $X$ and $Y$.
2. b) Calculate the centre of gravity of distribution $(\bar{x}, \bar{y})$.
3. c) Calculate the covariance $\sigma_{xy}$.

Analytically prove that if the empirical relationship between two variables $X$ and $Y$ is exactly linear and satisfies $y_i = a x_i + b$ for all pairs of values (with $a > 0$), then the covariance is $\sigma_{xy} = a \cdot \sigma_x^2$ and the linear correlation coefficient is exactly $r = 1$.

(Hint: recall that if $y_i = a x_i + b$, then the mean satisfies $\bar{y} = a \bar{x} + b$. Substitute these expressions into the covariance formula $\sigma_{xy} = \frac{\sum (x_i - \bar{x})(y_i - \bar{y})}{N}$).

The equations of the two regression lines for a bivariate distribution are known:

Line of $Y$ on $X$: $y = -1,2x + 10$

Line of $X$ on $Y$: $x = -0,6y + 8$

1. a) Calculate the coordinates of the centre of gravity $(\bar{x}, \bar{y})$.
2. b) Knowing that the correlation coefficient $r$ has the same sign as the slope of the regression lines, and that the relationship $r^2 = m_{yx} \cdot m_{xy}$ holds (where $m_{yx}$ is the slope of the line of $Y$ on $X$, and $m_{xy}$ is the slope of the line of $X$ on $Y$), calculate the exact value of $r$.

The covariance between variable $X$ (mean daily temperature in degrees Celsius) and variable $Y$ (soft drink sales in euros) in a given month is $\sigma_{xy} = 150$.

If we decide to change the temperature unit to degrees Fahrenheit using the conversion formula $F = 1,8C + 32$, answer with reasoning:

1. a) What will be the new value of the covariance $\sigma_{Fy}$ between temperature in Fahrenheit and sales?
2. b) Will the new linear correlation coefficient change relative to the original? Justify your answer referring to the formula for $r$.

A farmer wants to predict the yield of his tomato crop ($Y$, in tonnes). He has two possible predictor variables: the amount of fertiliser used ($X_1$, in kg) and the number of rainy days during cultivation ($X_2$).

After analysing historical data, he obtains the following parameters:

$\sigma_{x_1} = 10$, $\sigma_{x_2} = 4$, $\sigma_y = 5$.

Covariance between $X_1$ and $Y$: $\sigma_{x_1y} = 40$.

Covariance between $X_2$ and $Y$: $\sigma_{x_2y} = -18$.

1. a) Calculate the linear correlation coefficient for each pair of variables: $(X_1, Y)$ and $(X_2, Y)$.
2. b) Which variable provides a more reliable prediction of tomato yield using a linear model? Justify your decision.

In a bivariate distribution, it is known that the centre of gravity is the point $(4, 7)$, the variance of the variable $X$ is $\sigma_x^2 = 3$ and the equation of the regression line of $Y$ on $X$ is $y = 1,5x + 1$.

1. a) Verify algebraically whether the centre of gravity lies on the given line.
2. b) From the gradient of the regression line, calculate the value of the covariance $\sigma_{xy}$.
3. c) If it is also known that the variance of $Y$ is $\sigma_y^2 = 8$, what is the value of the linear correlation coefficient $r$?

At a secondary school, the relationship between the average grade obtained in $4.º\text{ de ESO}$ ($X$) and the average grade obtained in Bachillerato ($Y$) is analysed for a representative sample of students. The distribution parameters are:

$\bar{x} = 6,5$; $\bar{y} = 6,0$; $\sigma_x = 1,5$; $\sigma_y = 1,2$; $\sigma_{xy} = 1,44$.

The teaching staff needs to address two tasks:

1. Predict the Bachillerato grade of a female student who finished $4.º\text{ de ESO}$ with an $8,0$.

2. For a student who has obtained a $7,5$ in Bachillerato, estimate what average grade they likely obtained in $4.º\text{ de ESO}$.

To solve task 2, two different approaches are proposed:

• Method A: Algebraically isolate the unknown $x$ in the regression line of $Y$ on $X$ calculated in task 1, substituting $y = 7,5$.

• Method B: Calculate the regression line of $X$ on $Y$ and directly evaluate the value $y = 7,5$ in it.

1. a) Solve task 1 by calculating the appropriate regression line and providing the prediction for the student.
2. b) Apply both methods (A and B) for task 2 and compare the two numerical values they yield.
3. c) Justify which of the two methods is mathematically correct from the perspective of the method of least squares and reason under what single condition both methods would yield exactly the same result.

Carlos presents a project on the performance of two types of fuel ($X$ and $Y$) and claims to have obtained the following two regression lines:

• Regression line of $Y$ on $X$: $y = 2,5x - 3$

• Regression line of $X$ on $Y$: $x = 0,8y + 2$

In his final presentation, he confidently makes three statements:

1. "Since both lines have positive slopes, the data show a very strong positive linear correlation."

2. "The centroid of the sample is the point $(1; -0,5)$."

3. "If we change units by multiplying both variables by $10$, the new correlation coefficient will be multiplied by $100$."

Show in detail that Carlos's results are mathematically impossible by analysing the following three aspects:

1. a) The mathematical relationship between the slopes of both lines and the linear correlation coefficient $r$, deducing why these two lines cannot correspond to any real statistical distribution.
2. b) The geometric property that necessarily links the centroid $(\bar{x}, \bar{y})$ to both regression lines, checking whether the point proposed by Carlos satisfies it.
3. c) The invariance property of the correlation coefficient $r$ under positive linear scale changes.

Topic Challenge. Does the mobile phone steal hours from our night's rest? To find out, you will prepare a short statistical report using real data from your environment with the help of a spreadsheet or GeoGebra.

1. Data collection: Ask 15 family members or peers about their daily hours of screen time ($X$) and their hours of sleep ($Y$) from the previous day (they can check it in their device settings).

2. One-dimensional analysis: With variable $X$, build a grouped or ungrouped frequency table. Calculate the mean ($\bar{x}$), the variance ($s^2$), the standard deviation ($s$), and the quartiles ($Q_1$, $Me$, $Q_3$). Generate its box plot and interpret the dispersion and possible outliers.

3. Two-dimensional study: Plot the scatter plot for $(X, Y)$. Draw the regression line by eye or with the software and assess the sign and strength of the existing correlation.

4. Final report: Summarize your findings on one page, attaching the graphs and proposing a healthy recommendation based on the evidence obtained.

Assessment criteria:

• Precision in calculating measures of central tendency and dispersion, and in drawing the box plot.

• Accuracy in interpreting the two-dimensional relationship and the correlation between both variables.

• Autonomous use of GeoGebra or the spreadsheet and argumentative clarity in the report.