Confidence intervals with coefficient
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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" that fraternity threw last year. They took a sample of 52 fraternities on campus, resulting in the regression output below.
Find the 95% confidence interval for the true slope of βParties.
| Clue | Insight |
|---|---|
| We need to find the 95% confidence interval. | We're working with confidence intervals. |
| We're being asked to find the "true slope of βParties". | We're working with coefficients. (βParties is the population coefficient... we're just given the sample coefficient of bParties.) |
| We're given the sample coefficient (bParties), not the population coefficient (βParties). | We'll need to settle for t-scores (instead of z-scores). |
| Assumption | Validate |
|---|---|
| Linearity | For the sake of this example, let's assume the underlying scatterplot shows a linear relationship. ✅ |
| Independence | We can assume that each chapter's parties thrown and recruits received don't impact one another. ✅ (Ex: Delta Apple Pi's parties and recruits don't impact Alpha Blueberry Pi's.) |
| Equal Variance | For the sake of this example, let's assume the residual plot shows equal variance. ✅ |
| Normality | For the sake of this example, let's assume the residuals are normally distributed. ✅ |
b1 is the sample slope of the predictor variable.
t* is our critical value.
SE(b1) is the Standard Error (SE) for b1.
b1 = 20.5313
t* = ???
SE(b1) = 4.16456
To find our critical value (t*n-1), we'll need...
df = 50
α = (1 - Confidence Level) / 2
α = (1 - 0.95) / 2
α = (0.05) / 2
α = 0.025
t* = 2.009
CI = 20.5313 ± 2.009(4.16456)
CI = 20.5313 ± 8.36660104
CI = (12.165, 28.898)
Answer: We are 95% confident that the true increase in "Number of Recruits" for each "Party" (βParties) is between 12.165 and 28.898.
What if I have multiple coefficients?
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" that fraternity threw last year and the average "GPA" of the fraternity. They took a sample of 52 fraternities on campus, resulting in the regression output below.
Find the 95% confidence interval for the true slope of βGPA.
We'd run through the exact same process above, except zone in on bGPA instead of bParties!
