What is simple linear regression?
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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!
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!
