Hypothesis test 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.
Is there a meaningful, linear relationship between "Number of Recruits" and "Parties"? Provide support for your claim using a hypothesis test with an alpha level of 0.05.
| Clue | Insight |
|---|---|
| We're supporting our claim with a "hypothesis test". | We're working with hypothesis tests. |
| The claim is whether or not "there['s] meaningful, linear relationship between 'Number of Recruits' and 'Parties'", which is represented through βParties. | We're working with coefficients. |
| We're working with bParties (sample)... not the βParties (population). | We'll have to settle for a t-test (we can't take a z-test). |
| 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. ✅ |
H0: βParties = 0
Ha: βParties ≠ 0
It's in the regression output!
t* = 4.93
It's in the regression output!
p-value < 0.0001
α = 0.05
Considering our p-value is less than 0.0001 which is less than our alpha level of 0.05, this means we'll reject the null hypothesis!
Answer: Since our p-value is < 0.0001 is less than our alpha level of 0.05, we reject the null hypothesis and do have enough evidence to support the alternative hypothesis, which states that there is a meaningful linear relationship between "Number of Recruits" and "Parties" thrown by a fraternity at Crammer Nation University.
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.
Is there a meaningful, linear relationship between "Number of Recruits" and "GPA"? Provide support for your claim using a hypothesis test with an alpha level of 0.05.
We'd run through the exact same process above, except zone in on bGPA instead of bParties!
