For Questions 1-11, consider Model 1 below (Number of Recruits vs. Parties Thrown by Fraternity over the previous year).
Context: Crammer Nation University is wanting to develop a regression equation to predict the number of recruits a given fraternity will receive this rush season.
Model 1
Question #1: Write the simple regression equation for Number of Recruits vs. Parties Thrown by Chapter (over the previous year).
Answer:
Let's start with the basic formula for regression here, and fill in the values till we get to where we want.
To start, how can we identify what b0 should be?
Well, this is the y-intercept of the regression equation. It's the number of recruits a fraternity is predicted to get when they throw zero parties over the past year.
We can find this value here within our JMP output:
Therefore, we can place it in our regression equation like so:
Next, how can we find b1?
This value essentially answers the question: "For each party that a fraternity throws, how many more recruits will they receive?"
We can find that value here in our JMP output:
...which can be placed into our regression equation like so:
And that's it! That's our final regression equation for Number of Recruits vs. Parties Thrown by Chapter (over the previous year) based on the given JMP output!
Question #2: Describe the meaning of the y-intercept in this regression equation?
Answer: It's the number of recruits (85.8312) that a fraternity will be predicted to receive if the previous year they threw zero parties.
Mathematically speaking, in Question #1, we can see the y-intercept represented here:
This is the value of y-hat when x equals zero.
Conceptually speaking, the y-intercept is the point on the equation where x = 0. It's the value of the response variable (number of recruits) when the explanatory variable(s) (number of parties thrown in previous year) is zero.
That's why the y-intercept represents the number of recruits predicted for a given fraternity when they threw zero parties over the previous year!
Question #3: Describe the meaning of the slope of this regression equation?
Answer: With each party that a fraternity threw over the previous year, they're predicted to gain 20.5313 recruits.
Mathematically speaking, in Question #1, we can see the slope of the regression equation represented here:
With each unit increase in x (number of parties thrown over previous year), our number of recruits predicted increases by 20.5313.
For example, if a fraternity threw 0 parties, they'd be predicted to receive 85.8312 recruits.
If they threw 1 party, they'd be predicted to receive 85.8312 + 20.5313 = 106.3625 recruits.
If they threw 2 parties, they'd be predicted to receive 85.8312 + 40.0626 = 125.8938 recruits.
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