Linear correlation coefficient

In a study on solar panel efficiency, it has been calculated that the covariance between received solar radiation ($X$) and generated energy ($Y$) is $\sigma_{xy} = 15,2$. Knowing that the standard deviations are $\sigma_x = 4,0$ and $\sigma_y = 3,9$, calculate Pearson's linear correlation coefficient and interpret the result.

Classify the strength (strong, moderate, or weak/none) and direction (positive or negative) of the linear correlation for each of the following coefficients obtained in different statistical studies:

1. a) $r = 0,98$
2. b) $r = -0,12$
3. c) $r = -0,85$
4. d) $r = 0,55$

In an agronomy study, the linear correlation coefficient between the mean monthly temperature in degrees Celsius ($C$) and crop yield in tonnes ($P$) was calculated to be $r = 0,82$. If we decide to express the temperature in degrees Fahrenheit using the formula $F = 1,8C + 32$ and the yield in kilograms using $K = 1000P$, what will be the new correlation coefficient between $F$ and $K$? Justify your answer based on the mathematical properties of this coefficient.

Given two statistical variables $X$ and $Y$, their corresponding standardised variables are defined as $Z_x = \frac{X - \bar{x}}{\sigma_x}$ and $Z_y = \frac{Y - \bar{y}}{\sigma_y}$. Prove step by step that the covariance between the standardised variables, $\sigma_{z_x z_y}$, is exactly equal to the linear correlation coefficient $r_{xy}$ of the original variables.

Consider a bivariate distribution $(X, Y)$ where the variable $Y$ is obtained from the variable $X$ by means of the affine transformation $Y = -4X + 7$, with $X$ not constant ($\sigma_x > 0$). Prove analytically, using the algebraic definitions of covariance and standard deviation, that the linear correlation coefficient between $X$ and $Y$ is exactly $-1$.

The following scatter diagram represents the relationship between the temperature of a chemical reactor ($X$, centred at 0) and the reaction yield ($Y$). The recorded points are $(-2, 4)$, $(-1, 1)$, $(0, 0)$, $(1, 1)$, and $(2, 4)$.

1. a) Calculate Pearson's linear correlation coefficient for these data.
2. b) In light of the mathematical result and the attached diagram, explain carefully why the correlation coefficient takes that value even though an exact functional relationship ($y = x^2$) exists between the variables.

A basketball coach analyses the minutes played ($X$) and the points scored ($Y$) by 6 of his players in a match, obtaining the following pairs of data: $(10, 4)$, $(15, 8)$, $(20, 12)$, $(25, 14)$, $(30, 20)$, $(35, 22)$. Calculate the coefficient of determination $R^2$ and explain what percentage of the variability of points scored is explained linearly by minutes played.

The coefficient of determination of a mathematical model relating the age of a commercial vehicle ($X$, in years) to its market value ($Y$, in euros) is $R^2 = 0,7225$. Knowing that as age increases, the value of the vehicle decreases, what is the exact value of the linear correlation coefficient $r$?