Simple Linear Regression

9. Simple Linear Regression: Simple Linear Regression

Simple Linear Regression

The most basic form of regression analysis is simple linear regression.
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Simple linear regression is a statistical technique used to model the linear relationship between two variables.

It does this by finding the best-fitting straight line through a set of data points.
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This best-fitting line is called the regression line and is mathematically described by the following regression equation:

$\hat{Y} = b_0 + b_1X$

where $\hat{Y}$ is the predicted value of $Y$ .

Usefulness of the Regression Equation

Finding the values of the coefficients of a regression equation serves a dual purpose:

It helps us understand the relationship between the two variables. The slope of the equation, for instance, allows us to determine the direction and strength of the linear relationship between $X$ and $Y$ .
It allows us to make predictions about $Y$ on the basis of $X$ . Once we have found the regression equation, we can simply plug in a value for $X$ to make a prediction about the value of $Y$ .

For $10$ days, the owner of an ice cream truck kept track of how much ice cream he sold and what the maximum temperature in $^\circ{}C$ was that day. He then performed a simple linear regression to construct a regression line in the hopes of finding the relationship between the maximum temperature and the amount of ice cream sold.

Take a look at the scatterplot below. The blue dots represent the $10$ $\blue{\textbf{data points}}$ that serve as the basis for the regression analysis. The $\orange{\textbf{regression line}}$ $\hat{Y} =-20.45 + 2.93X$ is drawn in orange.
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The slope $b_1$ is $2.93$ . This value predicts how much more ice cream $Y$ will be sold, given that the maximum temperature $X$ increases by $1$ . For example, if the maximum temperature increases by $2$ , the amount of ice cream sold is predicted to increase by $2\cdot 2.93=5.86$ .

The intercept $b_0$ is $-20.45$ . In this case, the negative value of the intercept holds no particular meaning, since it not possible to sell a negative amount of ice cream.

To calculate the predicted amount of ice cream sold at a particular maximum temperature, we simply enter a value for $X$ into the equation. For example, at a maximum temperature of $X=25$ , the predicted amount of ice cream sold is:

$\hat{Y}=-20.45 + 2.93X=-20.45 + 2.93\cdot25=52.8$