ECON 222

1.    (4 marks) Simplify the following expressions.

a.    Find

b.    Find , where

2.    (4 marks) The population mean and variance of the random variable X are  and , respectively.  Prove that, for a sufficiently large sample size  n   and  the sample mean

a.   

b.   

3.    (6 marks) Let X be a random variable with a probability density function (PDF) given by

a.    Solve for c.

b.    Calculate .

c.    Calculate .

4.    (2 marks) A random sample of n voters is selected to estimate the proportion of voters who plan to vote for Candidate A in an election, .  How large does n need to be so that we can obtain a 90 percent confidence interval with a margin of error of .

5.    (10 marks) Let X and Y be two continuous variables with a joint PDF given by

a.    Calculate the marginal PDF of X.

b.    Calculate the marginal PDF of Y.

c.    Briefly explain if X and Y are independent.

d.   Calculate .

e.    Calculate .

6.    (10 marks) The data file assignment.xlsx contains the grades for 33 students on assignment 1  and assignment 2 .  Let  be normally and independently distributed with a mean and variance of  and , respectively.

a.    Calculate .

b.    Calculate .

c.    State the appropriate null and alternative hypotheses to test whether the performance on the assignments does not differ.

d.   Briefly explain whether a t- or Z-test is more appropriate.

e.    Perform the appropriate test at the 5-percent level of significance and briefly explain your conclusion.

7.    (10 marks) The data file fultonfish.dat shows the daily sales of fish (in pounds) for a period of time.

a.    Test  against  at the 5-percent level of significance.  Briefly explain your result using a diagram showing the estimated value of the test statistic and the critical value.

b.    Calculate the p-value of the test statistic and briefly explain how it can be used to perform the hypothesis test.  Show the p-value in the diagram.

Let total weekly sales be given by , where  represents sales on weekday i.

c.    Calculate.

d.   Calculate.

e.    Derive the probability distribution of  and calculate a 95-percent confidence interval estimate for .

8.    (6 marks) A police chief claims that the standard deviation in the length of response times is less than 3.7 minutes. A random sample of 9 response times from a normal population has a standard deviation of 3.0 minutes.

a.         State the appropriate null and alternative hypotheses.

b.         Briefly explain whether a t- or -statistic is more appropriate.

c.         Perform the appropriate test for  and briefly explain your conclusion.