Stat 552
# Homework 2

##### Due 3pm Jan 22 in class

- Faraway Chapter 2.1-2.7
- Model Formula in R : Chapter 11.1 in Introduction to R
- The recycling rule

Hardcopy handed in (may be handwritten or typeset)

**1.** Consider the simple linear regression model:

where the are independent Normal random errors.

a) Using the matrix form for multiple linear regression, write out the form of , and .

b) Calculate , and .

c) Find the least squares estimates, .

d) (Extra credit) Show the least squares estimates above, are equivalent to the usual form for the estimates in simple linear regression:

**2.** Using the matrix from of the least squares estimates, derive the form of the estimate of slope in a simple linear regression without an intercept:

where the are independent Normal random errors.

Hand in a .Rmd file on canvas, and as a hardcopy of the compiled .Rmd file (i.e. a .pdf of .doc).

**1.** The dataset `teengamb`

in the package `faraway`

contains survey data from a study to investigate teen gambling in the U.K.

Consider the regression model:

a) Construct the design matrix and response vector in R.

b) Find the least squares estimates using matrix algebra in R. Verify your answers by fitting the regression model using `lm`

.

c) Find the fitted values and residuals using matrix algebra in R and present a plot of residuals against fitted values.

**2.** Consider the following regression model:

Simulate a realization of the model in R by setting up the matrix, vector, and the error vector and using matrix algebra. Include a plot of against .

**3.** (Faraway 2.3) Generate some artifical data by:

Fit a polynomial in `x`

for predicting `y`

. Compute in two ways: using `lm`

, and directly using matrix algebra. At what degree of polynomial does the direct calculation fail?

A regression model that uses a polynomial in of degree to predict is

(**Take away:** while the matrix formulas we use in class are analytically correct, they donâ€™t neccessarily describe a good way to get answers numerically.)