Parameters and the regression line

An online film archive wants to check whether there is a relationship between the number of instalments in a film franchise ($X$) and the average viewers' rating (out of 10 points, $Y$). The following parameters are available after analysing 50 franchises:

$\sum x_i = 150 \quad \sum y_i = 325 \quad \sum x_i^2 = 580 \quad \sum y_i^2 = 2200 \quad \sum x_i y_i = 940$

1. a) Calculate the covariance $\sigma_{xy}$.
2. b) Interpret the result obtained in the context of the problem: do longer franchises tend to have better or worse ratings?

The 'centre of gravity' of a bivariate distribution is the point formed by the means of both variables, denoted as $(\bar{x}, \bar{y})$. A group of friends recorded the time they spend on social media per day ($X$, in minutes) and the hours they sleep each night ($Y$, in hours):

$(45, 9), (60, 8,5), (90, 8), (120, 7), (150, 6,5)$

Calculate the coordinates of the centre of gravity of this distribution.

A sociological study analyses the relationship between the number of books read per year ($X$) and the score on a mental agility test ($Y$). The covariance is $12,5$. If the number of books read by all respondents is doubled and 5 points are added to everyone's score on the mental agility test:

1. a) Reason what happens to the new covariance $\sigma'_{xy}$.
2. b) If the original correlation coefficient was $r = 0,75$, what will the new coefficient $r'$ be after the changes? Justify your answer analytically.
3. c) Will the reliability of the predictions change after these modifications to the data?

In a study on the performance of e-scooters, the ambient temperature ($X$, in $^\circ C$) and the actual range achieved ($Y$, in km) are analysed. The following data are known:

• The variance of the temperature is $s_x^2 = 16$.

• The regression line of the range on the temperature is $y = -0,85x + 42$.

• The coefficient of determination is $R^2 = 0,7225$.

Calculate:

1. a) The covariance between the temperature and the range.
2. b) The standard deviation of the range, $s_y$.
3. c) What would happen to the correlation coefficient if we decided to measure the temperature in degrees Fahrenheit using the transformation $F = 1,8X + 32$? Justify your answer based on the properties of correlation.

Observe the following scatter diagram representing the relationship between the advertising expenditure of 8 small businesses and their monthly profits (both in thousands of euros).

1. a) Justify, without performing calculations, whether the sign of the covariance will be positive or negative.
2. b) Choose with reasoning which of the following values could be Pearson's correlation coefficient ($r$): $-0,85$; $0,12$; $0,91$; $1,5$.
3. c) What would happen to the value of $r$ if we decided to express the data in euros instead of thousands of euros?

A nutritionist studies the relationship between the grams of sugar ($X$) and the kilocalories ($Y$) of five types of energy bar. She has obtained the following marginal parameters:

• Mean sugar: $\bar{x} = 12$ g

• Mean energy: $\bar{y} = 210$ kcal

• Standard deviations: $\sigma_x = 3$ g and $\sigma_y = 25$ kcal

• Covariance: $\sigma_{xy} = 60$

Calculate the linear correlation coefficient ($r$) and interpret whether there is a strong relationship between both variables.

If the coefficient of determination of a bivariate distribution is $R^2 = 0,81$, calculate the value of the correlation coefficient $r$ (assuming the relationship is positive) and explain what percentage of the variation in variable $Y$ is explained by variable $X$.

A music streaming app analyses the relationship between the age of its users ($X$) and the number of different artists they listen to per month ($Y$). The following summations are known for a sample of $n = 10$ users:

1. Calculate Pearson's correlation coefficient.

2. Determine the coefficient of determination $R^2$ and interpret its meaning: what percentage of the variation in the number of artists listened to is explained by the user's age?

A farmer observes that the yield of his orange crop depends on the hours of sunshine received during the ripening stage. The regression lines obtained from historical data are:

Regression line of $Y$ on $X$: $y = 0,85x + 12$

Regression line of $X$ on $Y$: $x = 0,92y - 4$

1. a) Determine the centre of gravity $(\bar{x}, \bar{y})$ of the distribution.
2. b) Calculate Pearson's correlation coefficient from the gradients of the two regression lines.

For a two-variable distribution, it has been verified that the standard deviations of both variables are equal ($s_x = s_y$).

1. a) Prove algebraically that, in this case, the linear correlation coefficient coincides with the slope of the regression line of $Y$ on $X$.

1. b) If the regression line of $Y$ on $X$ is $y = 0,6x + 2$, what is the coefficient of determination?

1. c) What would be the slope of the regression line of $X$ on $Y$ under these conditions?

Given the regression lines $y = -2x + 10$ and $x = -0,4y + 4$, decide whether the following statements are true or false, proving your answer by calculating the corresponding parameters:

1. a) The correlation between the variables is positive because both slopes have the same sign.

1. b) The centroid of the distribution is the point $(3, 4)$.

1. c) The variables have a perfect functional dependence.

1. d) The variance of $Y$ is greater than the variance of $X$.