Bivariate descriptive statistics

When studying relationships between two variables, it is essential to distinguish whether there is an exact dependence or a statistical relationship. State in which of the following cases the relationship is functional (an exact formula exists) and in which it is statistical (a trend exists, but with variability):

• a) The number of litres of petrol refuelled at a petrol station and the total cost to pay.

• b) The mark obtained in a Mathematics exam and the number of hours the student spent studying that week.

• c) The time an energy-saving light bulb remains switched on and the energy consumption recorded by the meter.

• d) A person's height and their shoe size.

• e) The maximum daily temperature in a city and the number of ice creams sold at a kiosk in that same city.

In a study on healthy habits, the distance cycled per week in km ($X$) and bicycle maintenance expenditure in euros ($Y$) are measured. It is known that the regression line of $Y$ on $X$ is $y = 0.12x + 15$ and that the correlation coefficient is $r = 0.85$.

1. State whether the following statements are true or false, justifying your answer:

   • a) There is a functional dependence between km cycled and expenditure.

   • b) The relationship between the variables is direct.

   • c) If a cyclist cycles 0 km ($x=0$), the model estimates an expenditure of $15$ €.

2. If the average annual expenditure is $\bar{y} = 33$ €, what is the mean weekly km cycled $\bar{x}$?

A group of 20 teenagers has been classified according to their reading habit (Frequent reader / Non-reader) and their preference for book format (Paper / Digital). The data obtained are:

• 8 are frequent readers and prefer paper format.

• 4 are frequent readers and prefer digital format.

• 2 are non-readers but prefer paper format.

• 6 are non-readers and prefer digital format.

Organise this information into a joint frequency table (contingency table), including the marginal totals in the rows and columns.

From the following contingency table showing the main type of device used to study and the student's school year (3rd Year or 5th Year), answer the questions asked:

• a) What is the joint absolute frequency of students who are in 5th Year and mainly use a laptop?

• b) Determine the marginal frequency of students who use a tablet.

• c) How many students in total participated in the study?

• d) What percentage of all students are from 3rd Year?

Given the following joint frequency table for variables $X$ (number of children) and $Y$ (number of rooms in the dwelling):

1. a) Calculate the marginal distributions of $X$ and $Y$.
2. b) Determine the mean and variance of variable $Y$ conditioned on $X = 1$.
3. c) Calculate the covariance $S_{xy}$ and state whether the variables are independent or show some type of association.

A courier company studies the relationship between package weight in kg ($X$) and shipping cost in euros ($Y$). Five shipments have been analysed with the following results:

Calculate:

1. The centre of gravity of the distribution $(\bar{x}, \bar{y})$.

2. The marginal standard deviations $\sigma_x$ and $\sigma_y$.

The relationship between weekly study time ($X$, in hours) and the average mark ($Y$) of a group of students is analysed. It is known that the covariance is $S_{xy} = 4.5$ and the correlation coefficient is $r = 0.75$. If all students are rewarded with 1 extra mark on their average mark and, in addition, decide to study twice as many hours the following week:

1. a) Prove mathematically how the new covariance $S'_{xy}$ and the new correlation coefficient $r'$ vary.
2. b) Would the reliability of the linear model's predictions change after these changes? Justify your answer based on the properties of bivariate parameters.

Match each of the following values of Pearson's linear correlation coefficient, $r$, with the graph that best represents it:

• 1) $r = 0.95$

• 2) $r = -0.85$

• 3) $r = 0.05$

Given the following pairs of variables, state whether you expect to find a positive, negative or zero correlation:

• a) The number of kilometres driven by a car and the amount of fuel remaining in the tank.

• b) The average mark in Baccalaureate and the number of hours a student spends playing video games.

• c) A person's age and their ID number.

• d) A person's monthly income and the amount of money they save per month.

• e) An athlete's average speed in a race and the time taken to complete it.

A basketball club wants to predict the number of points scored by its players ($Y$) based on the minutes played ($X$). After analysing the players, the following data are obtained:

$\sum x_i = 240$

$\sum y_i = 120$

$\sum x_i^2 = 6500$

$\sum y_i^2 = 1800$

$\sum x_i y_i = 3200$

Number of players analysed: $n = 10$.

1. Calculate Pearson's linear correlation coefficient.

2. If a player plays $25$ minutes, estimate the points they will score using the corresponding regression line.

The regression line relating the daily usage time of a social network in minutes ($X$) to the battery level consumed in percentage ($Y$) has been obtained. The equation of the line is:

$y = 0.15x + 2.5$

• a) Estimate the percentage of battery consumed if a user uses the social network for $120$ minutes.

• b) If the linear correlation coefficient is $r = 0.92$, do you think the previous estimation is reliable? Justify your answer.

We have a bivariate variable $(X, Y)$ where $X$ represents the outdoor temperature in $^\circ\text{C}$ and $Y$ the electricity consumption of a shopping centre in $kW$. We know that the mean centre of the distribution is the point $(\bar{x}, \bar{y}) = (22, 450)$.

If the regression line of $Y$ on $X$ has a gradient of $b = 15$:

• a) Write the equation of the regression line of $Y$ on $X$ using the point-gradient form: $y - \bar{y} = b(x - \bar{x})$.

• b) What would be the expected consumption for a temperature of $30^\circ\text{C}$?

The following table shows the marks obtained by 6 students in a Mathematics exam ($X$) and an Economics exam ($Y$):

1. Find the covariance $\sigma_{xy}$.

2. Obtain the equation of the regression line of Economics ($Y$) on Mathematics ($X$).

3. What mark would be expected in Economics for a student who scored $10$ in Mathematics?

A sociological study relates a household's digitalization index ($X$, scale 1-10) to its annual savings ($Y$, in thousands of euros). It is known that the regression line of $Y$ on $X$ is $y = -1.2x + 15$ and that the coefficient of determination is $R^2 = 0.81$.

1. a) If the variance of the digitalization indices is $\sigma_x^2 = 4$, what is the variance of the savings variable $\sigma_y^2$?

1. b) If a household has a digitalization index of 8, estimate its annual savings and calculate the standard error of the estimate (residual standard deviation), defined as $\sigma_{res} = \sigma_y \sqrt{1 - R^2}$.

The regression lines of a bivariate distribution are $4x - 5y + 12 = 0$ and $9x - 10y + 15 = 0$.

1. a) Identify which of them is the regression line of $Y$ on $X$ and which is that of $X$ on $Y$. Justify your answer using the fact that the product of their gradients must equal $r^2$ and that $0 \leq r^2 \leq 1$.
2. b) Calculate the linear correlation coefficient $r$.
3. c) Determine the mean values $\bar{x}$ and $\bar{y}$.