Confidence intervals with coefficient

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.

ClueInsight
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.
Parties 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).
AssumptionValidate
LinearityFor the sake of this example, let's assume the underlying scatterplot shows a linear relationship. ✅
IndependenceWe 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 VarianceFor the sake of this example, let's assume the residual plot shows equal variance. ✅
NormalityFor 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

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!

Activate AutoScroll