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 $100$-meter 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.

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

$(40, 60)$, $(50; 80)$, $(60, 80)$, $(80, 120)$, $(90; 140)$, $(100, 160)$.

1. a) Plot the scatter plot 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 $(70; 106)$? Justify your answer by calculating the center of gravity 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 center of gravity $(\bar{x}, \bar{y})$ of this bivariate distribution.

In a 4th ESO class, the relationship between Physics marks ($X$) and Mathematics marks ($Y$) of the $20$ enrolled students was studied. The center 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 center 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).

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.

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 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 center of gravity of distribution $(\bar{x}, \bar{y})$.
3. c) Calculate the covariance $\sigma_{xy}$.

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 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 market study on house prices ($Y$, in thousands of pesos) based on floor area ($X$, in square meters), 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 covariance between variable $X$ (mean daily temperature in degrees Celsius) and variable $Y$ (soft drink sales in pesos) 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 metric tons). 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.

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 $\sigma_{xy} = a \times \sigma_x^2$ 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 $\sigma_{xy} = \frac{\sum (x_i - \bar{x})(y_i - \bar{y})}{N}$).

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?

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 pesos):

• Center of gravity: $(240, 40)$.

• Standard deviation of $X$: $\sigma_x = 60$.

• 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 $y = mx + n$.

In a bivariate distribution, it is known that the center 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 center of gravity lies on the given line.
2. b) From the slope 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 $r$?

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 center of gravity $(\bar{x}, \bar{y})$. (Recall the geometric property relating both lines to the center 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 center 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 $r^2 = m_{yx} \times m_{xy}$ 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 used car market value in thousands of pesos ($Y$) is $y = -24x + 360$.

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 transportation company wants to analyze the relationship between the distance covered on a trip ($X$, in tens of km) and fuel consumption ($Y$, in liters). The data recorded for a sample of $5$ trucks 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 trip 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.