Confidence intervals with predicted mean response
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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 µRecruits for chapters who threw 3 parties last year, given the mean parties thrown last year is 4.5.
| Clue | Insight |
|---|---|
| We need to find the 95% confidence interval. | We're working with confidence intervals. |
| We're being asked to find the "true µRecruits for chapters who threw 3 parties last year". | We're working with the response variable (given a predictor variable value). |
| The RMSE is derived from the sample... not the population. | 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. ✅ |
y-hatv is the predicted response (based on the regression) at a given xv value.
t* is our critical value.
SE(µ-hatv) is the standard error for the population mean at xv.
Why are we not using SE(y-hatv)? ➡️ We want the standard error of the true population mean [a.k.a. SE(µ-hatv)]... as we're working with a confidence interval for that true population mean. We don't want the standard error of the predicted response [a.k.a. SE(y-hatv)]!
y-hatv = ???
t* = ???
SE(µ-hatv) = ???
We need to calculate all of these!
How to calculate y-hatv
How to locate t*
df = 50
α = (1 - Confidence Level) / 2
α = (1 - 0.95) / 2
α = (0.05) / 2
α = 0.025
t* = 2.009
How to calculate SE(µv)
SE(b1) is the standard error of our coefficient b1.
xv is the predictor variable value we're gathering the confidence interval of the response for.
x-bar is the mean of the x-values.
se is the standard deviation of the residuals (a.k.a. Root Mean Square Error (RMSE)).
n is the number of observations.
SE(b1) = 4.16456
xv = 3
x-bar = 4.5
se = 43.8180
n = 52
SE(µv) = √[(4.16456)2 * (3 - 4.5)2 + ((43.8180)2 / 52)]
SE(µv) = √[17.3435 * (-1.5)2 + (1920.017124 / 52)]
SE(µv) = √[17.3435 * 2.25 + 36.9234]
SE(µv) = √[39.022875 + 36.9234]
SE(µv) = √[75.946275]
SE(µv) = 8.7147
