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 x_{1} value ("Parties Thrown").
b_{0} is the y-intercept.
b_{1} is the change in predicted response ("Number of Recruits") for each unit +/- in x_{1} ("Parties Thrown").
x_{1} is the value for the first (i.e. "1") predictor variable.
b_{0} = 85.8312
b_{1} = 20.5313
y-hat = 85.8312 + 20.5313(x_{1})
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?
x_{1} = 3
y-hat = 85.8312 + 20.5313(x_{1})
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!
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.
b_{0} is our estimate (with our sample) of the actual y-intercept for the population.
b_{1} is our estimate (with our sample) of the actual change in predicted mean response for each unit +/- in our predictor variable.
µ_{y} is the population's mean response to given predictor variable values.
β_{0} is the actual y-intercept for the population's regression.
β_{1} 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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Your university wants to predict the “Attendance” at a student organization’s event based on the “Social Media Posts” made by that organization in the weeks leading up to the event. They sample 30 student organizations and their events over the semester, running the numbers through regression analysis.
Make the prediction for sign-ups when a student organization has posted 3 times over the past week.
Predicted Sign-Ups:
Your university wants to predict the “Sign-Ups” at a student organization receives at mega fair based on the social media “Posts” made by that organization throughout the semester. They sample 30 student organizations, resulting in the following regression analysis.
Select the choices that represent the population regression equation.
Your university wants to predict the “Attendance” at a student organization’s event based on the “Social Media Posts” made by that organization in the weeks leading up to the event. They sample 30 student organizations and their events over the semester, running the numbers through regression analysis.
Select the choices that represent the prediction regression equation.
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