Homework assignment, due mardi, 10 nov

1. Exercise 2.3: Demonstration of the Central Limit Theorem: let x = x1 + · · · + x20, the sum of 20 independent Uniform(0,1) random variables. In R, create 1000 simulations of x and plot their histogram. On the histogram, overlay a graph of the normal density function. Comment on any differences between the histogram and the curve.

2. Exercise 8.1: Fitting the wrong model: suppose you have 100 data points that arose from the following model: y = 3+0.1x1+0.5x2+error, with errors having a t distribution with mean 0, scale 5, and 4 degrees of freedom. We shall explore the implications of fitting a standard linear regression to these data.

(a) Simulate data from this model. For simplicity, suppose the values of x1 are simply the integers from 1 to 100, and that the values of x2 are random and equally likely to be 0 or 1. (In R, you can define x.1 <- 1:100, simulate x.2 using rbinom(), then create the linear predictor, and finally simulate the random errors in y using the rt() function.) Fit a linear regression (with normal errors) to these data and see if the 68% confidence intervals for the regression coefficients (for each, the estimates ±1 standard error) cover the true values.

(b) Put the above step in a loop and repeat 1000 times. Calculate the confidence
coverage for the 68% intervals for each of the three coefficients in the model.

(c) Repeat this simulation, but instead fit the model using t errors (see Exercise 6.6).