What is it?

• Linear Regression is a statistics method that shows the relationship between an independent variable and a dependent variables and allows us to predict outcomes.

• Independent Variable: The input variable. Usually denoted as \(\mathcal{x}\)

• Dependent Variable: The output that the input variable effects. Usually denoted as \(\mathcal{y}\)

Example: AS The study hours increase to do the study scores Real-world data rarely follows a perfect linear pattern. Linear regression allows us to find the line that best represents the overall trend.

• The Regression Equation: An equation that best models the relationship between the independent and dependent variables. It is represented by \(\hat{y} = b_0 + b_1 \mathcal{x}\) where \(b_0\) is the constant and \(b_1\) is the slope.

• First find the slope:

  \(b_1 = \frac{\sum (x_i - \bar{x})(y_i - \bar{y})}{\sum (x_i - \bar{x})^2}\)

• Then find the Y-intercept:

  \(b_0 = \bar{y} - b_1 \bar{x}\)

  Where \(\bar{x}\) is the mean of \(\mathcal{x}\) and \(\bar{y}\) is the mean of \(\mathcal{y}\).

In order to better understand the relationship and make conclusions we would need to find the regression line for this graph.

Find the mean of x variables \({\bar{x}}\)

\(\frac{ 1 + 2 + 3 + 4 + 5 + 6 + 7}{7} = 4\)

Find the mean of y variables \({\bar{y}}\)

\(\frac{ 58 + 67 + 72 + 74 + 88 + 85 + 97}{7}= 77.286\)

Use the values in the previously mentioned equations to find the slope and y-intercept

\(b_1 = \frac{\sum (x_i - 4)(y_i - 77.286)}{\sum (x_i - 4)^2}= \frac{169}{28} = 6.036\)

\(b_0 = 77.286 - 6.036(4)= 53.143\)

Then plug them into the equation

\(\hat{y} = 53.143 + 6.036 \mathcal{x}\)

Here is the linear regression line on the graph

with our linear regression line we can now predict a students exam score based on their hours of study. Say a student studied 4.5 hours then:

\(\mathcal{x}=4.5\)

\(\hat{y} = 53.143 + 6.036(4.5)=80.43\)%