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Department of Economics
Economics 615: Econometrics I
Mid-term Exam
Directions: You may neither give nor receive help on this exam. It is a closed book exam: no books, no notes, no nothing. You must do all questions. Point values are shown. In order to receive partial credit you must show your work.
1. (20 points) A continuous variable is alleged to have the following probability density function
EMBED Equation.3
A. Prove that this is indeed a probability density. That is, find b.
B. What is the expected value of the random variable u?
C. What is the variance of u?
2. (10 points) A certain gasoline producer sponsors a mileage economy test involving thirty (30) cars. The miles per gallon, x, recorded to the nearest gallon, are given below.
XFrequency158169187206A. Find the average miles per gallon.
B. Find the variance for this sample.
3. (20 points) Consider the experiment of rolling a fair die, and define y as the number of dots showing face up. Let the events A and B be defined as follows: A=y is an even number, and B=y<3.
A. Find P(A EMBED Equation.3 B).
B. Find P(AUB).
C. Are the events A and B statistically independent?
4. (20 points) The following sample of size five has been drawn from a uniform probability distribution on the interval [a, b].
.0025
0.387
1.17
0.701
1.646
Find the sample mean.
Find the sample variance.
Use the method of moments to compute estimates of a and b.
5. (15 points) At the 5% level, test the hypothesis that the starting salaries of college graduates working in the Chicago area is equal to the starting salaries of graduates working in the Philadelphia area. A random sample of 100 is taken from the Chicago area with the result that the sample mean is $10,250 with sample standard deviation of 180. A random sample of 60 in the Philadelphia area yields a mean of $10,150 and a sample standard deviation of 160.
6. (15 points) The random variable x has the following distribution:
EMBED Equation.3 , which you recognize as the Poisson Distribution. The mean and the variance are both equal to .
The following sample is drawn:
1,1,4,2,0,0,3,2,3,5,1,2,1,0,0
At the 5% level of test, carry out a Wald test of the hypothesis that =2 against the alternate that it is not equal to two.
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