What is simple linear regression?

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

Scenario: Crammer Nation University wants to develop a regression equation to predict the "Number of Recruits" a given fraternity will receive this rush season given the "Parties Thrown" by the fraternity last year. They take a random sample of 52 fraternities on campus, resulting in the regression output below.

y-hat is the predicted value ("Number of Recruits") given an x1 value ("Parties Thrown").
b0 is the y-intercept.
b1 is the change in predicted response ("Number of Recruits") for each unit +/- in x1 ("Parties Thrown").
x1 is the value for the first (i.e. "1") predictor variable.

b0 = 85.8312
b1 = 20.5313

y-hat = 85.8312 + 20.5313(x1)

Now we can predict responses!

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

y-hat = 85.8312 + 20.5313(x1)
y-hat = 85.8312 + 20.5313(3)
y-hat = 85.8312 + 61.5939
y-hat = 147.4251

Answer: If Delta Apple Pi threw 3 parties last year, we can predict they'll receive 147.4251 recruits this rush season!

Population regression vs. Sample regression

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

y-hat is our estimate of what the actual mean response (µy) is at given predictor values.
b0 is our estimate (with our sample) of the actual y-intercept for the population.
b1 is our estimate (with our sample) of the actual change in predicted mean response for each unit +/- in our predictor variable.

µy 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 variable.

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!

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