What is multiple linear regression?

Multiple Linear Regression creates a line of best fit (a.k.a. "regression line") to represent the relationship between multiple predictor variable(s) and a response variable.

We have multiple predictor variables!

y-hat is the predicted value ("Number of Recruits") given predictor values x_k (x₁ = "Parties Thrown", x₂ = "GPA").
b₀ is the y-intercept.
b_k is the change in predicted response ("Number of Recruits") for each unit +/- in x_k (x₁ = "Parties Thrown", x₂ = "GPA").
x_k is the value for each of the predictor variables (x₁ = "Parties Thrown", x₂ = "GPA").

b₀ = 68.2912
b₁ = 22.6973
b₂ = 16.3921

y-hat = 68.2912 + 22.6973(x₁) + 16.3921(x₂)

Scenario: Delta Apple Pi threw 3 parties last year and had an average GPA of 3.2. Based on your linear regression, how many recruits do you predict they'll receive this rush season?

x₁ = 3
x₂ = 3.2

y-hat = 68.2912 + 22.6973(x₁) + 16.3921(x₂)
y-hat = 68.2912 + 22.6973(3) + 16.3921(3.2)
y-hat = 68.2912 + 68.0919 + 52.45472
y-hat = 188.83782

Answer: If Delta Apple Pi threw 3 parties last year and had an average GPA of 3.2, we can predict they'll receive 188.83782 recruits this rush season!

Our sample regression is our estimate of what the population regression actually equals!

y-hat is our estimate of what the actual mean response (µ_y) is at given predictor values.
b₀ is our estimate (with our sample) of the actual y-intercept for the population.
b_k is our estimate (with our sample) of the actual change in predicted mean mean response for each unit +/- in our predictor variables (x_k).

µ_y is the population's mean response to given predictor variable values.
β₀ is the actual y-intercept for the population's regression.
β₁ is the actual change in the population's mean response for each unit +/- in our predictor variables (X_k).

We can never know the true population regression... so we do our best and gather a sample from the population to develop a regression line!