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 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.

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 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.

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?

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 $r$.

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 $y = mx + n$.

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).

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 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?

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 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 ($X=18$).

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.

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.