Regression line of y on x
When a scatter plot shows a linear trend, it makes sense to summarise it with a line that allows us to predict. But there are two possible lines — predicting $Y$ from $X$, or $X$ from $Y$ — and in general they are different. Understanding why, and when to trust a prediction, is the aim of this section.
The regression line of $Y$ on $X$
Given the statistical parameters of a bivariate distribution: mean of $X$ equal to $4$, mean of $Y$ equal to $6$, variance of $X$ equal to $1,5$, and covariance equal to…
Given the statistical parameters of a bivariate distribution: mean of $X$ equal to $4$, mean of $Y$ equal to $6$, variance of $X$ equal to $1,5$, and covariance equal to $3$. Determine the equation of the regression line of $Y$ on $X$.
A bivariate distribution has as its data matrix the following summations: $\sum x_i = 20$, $\sum y_i = 40$, $\sum x_i^2 = 120$, $\sum x_i y_i = 180$, and the total…
A bivariate distribution has as its data matrix the following summations: $\sum x_i = 20$, $\sum y_i = 40$, $\sum x_i^2 = 120$, $\sum x_i y_i = 180$, and the total number of observations is $N = 10$. Calculate the covariance and the variance of $X$, and write the equation of the regression line of $Y$ on $X$.
The two lines and their relationship with $r$
The regression lines of a bivariate distribution are given by the equations $y = 0,5x + 4$ and $x = 1,6y - 2$. Determine the coordinates of the mean point of the…
The regression lines of a bivariate distribution are given by the equations $y = 0,5x + 4$ and $x = 1,6y - 2$. Determine the coordinates of the mean point of the distribution and the linear correlation coefficient $r$, giving reasons for its sign.
In a macroeconomic analysis, the inflation rate ($X$, in $\%$) is related to the unemployment rate ($Y$, in $\%$). The regression line of $Y$ on $X$ is $y = -0,6x + 8$.…
In a macroeconomic analysis, the inflation rate ($X$, in $\%$) is related to the unemployment rate ($Y$, in $\%$). The regression line of $Y$ on $X$ is $y = -0,6x + 8$. If the marginal variance of inflation is $\sigma_x^2 = 4$ and the marginal variance of unemployment is $\sigma_y^2 = 2,25$, determine the linear correlation coefficient $r$ and the percentage of the variability of unemployment that is explained by the linear regression model.
When $|r|=1$, the two lines coincide: proof
Prove analytically that if the variables $X$ and $Y$ have a perfect linear correlation ($r = 1$ or $r = -1$), then the regression line of $Y$ on $X$ coincides exactly…
Prove analytically that if the variables $X$ and $Y$ have a perfect linear correlation ($r = 1$ or $r = -1$), then the regression line of $Y$ on $X$ coincides exactly with the regression line of $X$ on $Y$. Briefly explain what geometric implication this has for the angle formed by both lines in the plane.
In a pharmaceutical study, the concentration of an active ingredient in blood ($Y$, in mg/L) is modelled as a function of the administered dose ($X$, in mg). If we…
In a pharmaceutical study, the concentration of an active ingredient in blood ($Y$, in mg/L) is modelled as a function of the administered dose ($X$, in mg). If we change variables by standardising both variables, that is, we define $Z_x = \frac{X - \bar{x}}{\sigma_x}$ and $Z_y = \frac{Y - \bar{y}}{\sigma_y}$, prove algebraically that the regression line of $Z_y$ on $Z_x$ passes through the origin and that its gradient is exactly the linear correlation coefficient $r$.
Predicting with caution: interpolation and extrapolation
In a thermodynamics experiment, the temperature of a liquid ($Y$, in ºC) is measured as a function of elapsed time ($X$, in minutes). The regression line obtained is…
In a thermodynamics experiment, the temperature of a liquid ($Y$, in ºC) is measured as a function of elapsed time ($X$, in minutes). The regression line obtained is $y = -1,5x + 85$. Knowing that the coefficient of determination is $R^2 = 0,95$ and that the total duration of the experiment was $30$ minutes, is it reliable to estimate the temperature at $45$ minutes using this line? Justify your answer based on the concepts of interpolation and extrapolation.
In a biomechanics study, the relationship between femur length ($X$, in cm) and height of an individual ($Y$, in cm) is analysed. The following statistical summaries are…
In a biomechanics study, the relationship between femur length ($X$, in cm) and height of an individual ($Y$, in cm) is analysed. The following statistical summaries are available for a sample of $50$ people:
$\bar{x} = 45$, $\bar{y} = 170$, $\sigma_x = 3,2$, $\sigma_y = 8,5$, $r = 0,92$.
Find the regression line of $Y$ on $X$ and estimate the height of a person whose femur measures $48$ cm. Briefly justify the reliability of this estimate.