AE 03: Inference

NCAA Football Expenditures

Important

Go to the course GitHub organization and locate your ae-03 repo to get started.

Render, commit, and push your responses to GitHub by the end of class to submit your AE.

Set up

library(tidyverse)
library(tidymodels)
library(knitr)

football <- read_csv("data/ncaa-football-exp.csv")

Data

Regression model

exp_fit <- lm(total_exp_m ~ enrollment_th + type, data = football)

tidy(exp_fit)|> 
  kable(digits = 3)

term	estimate	std.error	statistic	p.value
(Intercept)	19.332	2.984	6.478	0
enrollment_th	0.780	0.110	7.074	0
typePublic	-13.226	3.153	-4.195	0

Hypothesis test

We want to conduct a hypothesis test to determine if there is a linear relationship between enrollment and football expenditures after accounting for institution type.

We’ll start by getting estimates for statistics we’ll need for inference.

Exercise 1

We will use the vector of responses \(\mathbf{y}\) and the design matrix \(\mathbf{X}\) to calculate the values needed for inference.

Get \(\mathbf{y}\) and \(\mathbf{X}\) from the football data frame. What are their dimensions?

Exercise 2

Next, let’s calculate \(\hat{\sigma}_\epsilon^2\) the estimated regression standard error. Use \(\mathbf{y}\) and \(\mathbf{X}\) from the previous exercise to calculate this value.

Exercise 3

Now we’re ready to conduct the hypothesis test between enrollment and football expenditures. State the null and alternative hypotheses in words and using mathematical notation.

Exercise 4

Calculate \(SE(\hat{\beta}_j)\), then use this value to calculate the test statistic for the hypothesis test.

Exercise 5

Now we need to calculate p-value to help make our final conclusion.

• State the distribution used to calculate the p-value.

• Fill in the code below to calculate the p-value. Remove #| eval: false once you’ve filled in the code.

2 * pt([test-statistic], [df], lower.tail = FALSE)

Exercise 6

State your conclusion in the context of the data. Use a threshold of \(\alpha = 0.05\).

Submission

To submit the AE:

Render the document to produce the PDF with all of your work from today’s class.

Push all your work to your AE repo on GitHub. You’re done! 🎉