This is an exam!! You can use any written material and online source, but you can NOT discuss the exam with ANYONE

Problem 1

Let \(X\sim N(1,1)\), \(Y\sim N(2,1)\), \(Z\sim N(3,1)\), all independent. Find

\[P\left((X+Z)^2/2+Y^2<10\right)\]

Problem 2

highways in Resma3 has data on the rate of accidents on a number of highways, together with a list of predictor variables. These are

rate: accidents per million vehicle miles
len: length of the segment in miles
adt: average daily traffic count in thousands
trks: truck volume as a percent of total traffic
slim: speed limit
lwid: lane width in feet
shld: with in feed of outer shoulder
itg: number of freeway-type interchanges per mile
sigs: number of signalized interchanges per mile
acpt: number of access points per mile
lane: total number of traffic lanes in both directions
fai: 1 if federal aid interstate highway, 0 otherwise
pa: 1 if principal arterial highway, 0 otherwise
ma: 1 if major arterial highway, 0 otherwise

Some of these predictors are indicator variables, ignore this issue here.

Do all the calculations without using the lm command

  1. Find the least squares estimators of the predictors

  2. Find the \(R^2\)

  3. Find an estimate of \(\sigma^2\)

  4. Test at the 5% level whether the 8 predictors that have the lowest correlation with rate can be eliminated from the model.

Problem 3

Prove theorem (6.5.1). Make sure that your proof is complete and references all theorems used.

Problem 4

Say we have a model of the form \(y_i = \alpha x_i+\beta z_i+\epsilon_i\), that is a no-intercept model with two predictor variables. Find the least squares estimators of \(\alpha\) and \(\beta\)

  1. directly

  2. using the theorems from class.

  3. Show that this model is a generalization of the simple regression model, and use the formulas found in a. to derive the least squares estimators of the simple regression model.