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 β_{Parties}.
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 β_{Parties}". | We're working with coefficients. (β_{Parties} is the population coefficient... we're just given the sample coefficient of b_{Parties}.) |
We're given the sample coefficient (b_{Parties}), not the population coefficient (β_{Parties}). | 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. ✅ |
b_{1} is the sample slope of the predictor variable.
t* is our critical value.
SE(b_{1}) is the Standard Error (SE) for b_{1}.
b_{1} = 20.5313
t* = ???
SE(b_{1}) = 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" (β_{Parties}) is between 12.165 and 28.898.
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 β_{GPA}.
We'd run through the exact same process above, except zone in on b_{GPA} instead of b_{Parties}!
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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 randomly sample 30 student organizations, resulting in the following regression analysis.
Find the 95% confidence interval for the true slope of β_{Posts.}
We are % confident that the true population parameter is between and .
(When calculating test statistic, round to intermediary values to 3 decimal places. Round bounds to 3 decimal places.)
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