Application Exercise

1. Open Rstudio Server Pro
2. Create a new project
3. Click File > New File > R Markdown
4. Delete everything except for

---
title: "Untitled"
output: html_document
---

5. Give your work a sensible title
6. When you’ve finished, upload the .html file to Canvas

Input the design matrix, \(\mathbf{X}\) from the following data in R

Estimate the \(\beta\) coefficients for predicting y from age.

We are interested in predicting a chicken’s weight based on their diet using the chickwts dataset

• Fit the model of interest and extract the estimated \(\beta\) coefficients
• Construct \(\mathbf{x}_0\) for a chicken that is eating “sunflower”.
• Find the predicted weight for a chicken eating sunflowers.

Show that the variance of \(\mathbf{x}_0^T\hat\beta\) is

\[\textrm{var}(\mathbf{x}_0^T\hat\beta) = \mathbf{x}_0^T(\mathbf{X}^T\mathbf{X})^{-1}\mathbf{x}_0\sigma^2\]

• What is the expected value of \(\mathbf{x}_0^T\beta + \epsilon\)?
• Show that the variance is

\[(1 + \mathbf{x}_0^T(\mathbf{X}^T\mathbf{X})^{-1}\mathbf{x}_0)\sigma^2\]