Temple University
Econometrics I
1. Bowls and Chips

(a) Probability of W

or .656 

(b) Probability of b1 given white chip

or 

2. For each function find c so that f(.) is a pdf
(a) x/c where x = 1, 2, 3, 4. The best way to proceed is to set up the sum of the probabilities for this discrete random variable

Now 10/c must equal 1 in order for this to be a pdf, therefore c = 10

(b) c(x+1)2 where x = 0, 1, 2, 3

Now 30c must equal one, so c = 1/30.
(c) x3/4 where x is on the interval 0<x<c 

Now c4/16 must equal one so c = 2 
3. Advertised weight

(a) Expected mean weight is 24.43 
(b) Variance of sample mean, Var(Xbar) = 2.2/30 =0.0733 
(c) Prob that sample mean is in interval?

4. Ten econometrics students
First the original data:

(a) Compute the least squares estimates

(b) Residual Sum of Squares for the model 

(c) Estimate of error variance

(d) t-test for slope

Do not reject the null.
5. A good starting point is the summary found in the online lecture notes at Summary
From those notes you can infer the diagram below and which leads us to the following two parts to our answer:
A. The Wald is based on the horizontal distance between bo and b-hat.
The LR is based on the vertical difference between the likelihood at bo and the likelihood at b-hat. The Langrange multiplier test is based on the slope of the likelihood at bo.

B. The Wald Statistic is computed using only the unrestricted sample statistic.
The LR requires that we compute the empirical likelihood at both the sample and at the null.
The LM test uses the likelihood evaluated only at the restriction.