Due: Beginning of lecture, January 22, 1998

 INSTRUCTIONS: These problems pertain to Chapters 3 and 4 in Gujarati. Some advance reading may be required to complete these problems, although the material is still a review of the prerequisite Economics 40 or the equivalent.

NETWORK FILES NEEDED: n:alaska.dat, n:alaska0.sha

1. Is either of the following a valid probability density function? Why or why not?

 , (this should be very familiar)

 2. For a standard normal random variable Z, what is Pr(Z = 0.5)? Don't be fooled. Explain.

 3. If Z is the standard normal random variable, use the tables at the back of the text to determine:

 a.) Pr (Z > 0.5)

b.) Pr (0.1 < Z < 0.9)

c.) Pr (-0.2 < Z < 1.4)

d.) Pr (|Z| > 1.96) (an important one!)
 

4. If X is approximately normally distributed with mean 6 and variance 9, determine

a.) Pr (X > 7)

b.) Pr (4 < X < 7)

c.) x* such that Pr (X < x*) = .025
 

5. Distinguish between a population "parameter," an "estimator," and an "estimate."

6. Distinguish between "point estimation" and "interval estimation."

7. Is the median a "linear estimator"? Is the mean?

8. Would you rather used a "biased estimator" that was somewhat more "efficient," or an "unbiased estimator" that was relatively less "efficient"? Discuss.
 

9. If we draw a random sample of size 36 and we find that the sample mean is 7 and the sample variance is s²=4, construct a 95% "confidence interval" for the value of the population mean m _x. Show your work carefully.

10. If we draw a random sample of size 25 and discover that the mean of X in the sample is 10 and s²=16, test the null hypothesis that m _x=8.5 (using either a "two-tailed" Z-test or a "two-tailed" t-test, whichever is most appropriate, justifying your choice):

a.) at the 5% "level of significance";

b.) at the 10% "level of significance".
 

 
11. EXPLORING A DATA SET (Calisthenics with SHAZAM): Download from the network (or from the website) to your own diskette the files n:alaska.dat and n:alaska0.sha using the procedure outlined in the SHAZAM computer software orientation handout. Be sure to print out a copy of the program itself so you can refer to it later.

A description of the data set is contained in comment lines at the top of the program file.

a.) Start the SHAZAM program and when it says "TYPE COMMAND" invoke an already-created set of commands from the alaska0.sha file by issuing the command "file 5 alaska0.sha." If you are working from a disk in the a: drive, use file 5 a:alaska0.sha.) Use the pause button to view intermediate steps; this run is simply to verify that you can read and use the data (i.e. that all files are in the right places). Enter the command stat to verify that you have all the data. Issue the command stop to exit the program. Where does the output from this run get sent?

b.) Now, use the TED editor (or any other editor that allows you to save the edited file as a DOS file) to make some changes to the alaska0.sha file. Instead of issuing commands interactively after executing the initial set of commands from the alaska0.sha file, incorporate some additional commands into the program. Until you think you have the program running smoothly and correctly, continue to have the output sent to the screen. When it all looks like it works fine, then divert the output to an output file by removing the comment character from the line at the top that reads file 6 alaska.out .

For example, look at the actual numbers and then produce a set of descriptive statistics for some of the variables in the augmented data set by using the commands:

print year ptot qtot rtot
stat ptot qtot rtot / pcor

c.) Descriptive statistics: What are the highest and lowest prices (in 1989 dollars) that have been observed over the 1964-1991 period? What has been the average size of the catch, in millions of pounds, over this time period? What has been the standard deviation in catch over this period?

d.) SOME ECONOMIC THEORY: When demand is "elastic" (such that a given percent change in price leads to a larger percent opposite change in quantity demanded), an increase in price results in a decrease in total revenues in a market. When demand is "inelastic" (such that a given percent change in price leads to a smaller percent opposite change in quantity demanded), and increase in price results in an increase in total revenues. If we (erroneously) considered all five major types of Alaskan salmon to be sold in one market, what does the correlation table produced by the / pcor option on the stat command imply about overall demand elasticity in this market? Would you consider this implication reliable? Why or why not? (Think about the implicit ceteris paribus requirement underlying "demand curves," i.e. that everything else be held constant.)

e.) Now take into account that there are five different "goods" involved--chum, king, pink, red, and silver salmon species, possibly each with a distinct market. Use stat / pcor and the crude plot option in SHAZAM to see whether catch levels for each of these five species "move together" over this time period. Comment.

stat cquant kquant rquant pquant squant / pcor
plot cquant year / nopretty
plot kquant year / nopretty
plot rquant year / nopretty
plot pquant year / nopretty
plot squant year / nopretty

IF you were to run this program with output directed to the screen, rather than to a file, you could easily use the more high-tech plotting options to plot multiple time series such as:

    plot cquant kquant rquant pquant squant year / ega line nopretty
or plot cquant kquant rquant pquant squant year / gnu line nopretty

If you have your own stand-alone computer equipment and an attached laser printer, you are welcome to experiment with the gnuplot options mentioned in the manual in an attempt to print out hard copies of graphics files. Note that these fancier files do not come out if you direct the output to a file. The cruder dot matrix plots will be typically be adequate for homeworks.

f.) Have the prices of these five species moved together? An appropriate stat output with interpretation will be sufficient to answer this question.

g.) Generate revenues from each species and assess whether these have moved similarly over the time period of these data. Use genr commands like:

genr crev=cquant*cprice
genr krev=kquant*kprice
genr rrev=rquant*rprice
genr prev=pquant*pprice
genr srev=squant*sprice
stat crev krev rrev prev srev / pcor

h.) If an Alaska commercial fisher targeted only one species, say the low-end chum salmon, would this fisher have felt much of a decrease in their income in 1989, assuming a fairly constant number of fishers and an even distribution of the catch? Comment. What about fishers who targetted king salmon? Note that you can limit all analysis to a subset of the observations, say observations 24 through 30 (1987-1993) by using the command sample 24 30. (To undo this limitation, issue the command sample 1 30.)

i.) OPTIONAL (If all the other tasks were easy for you): Try the GNUPLOT capability of SSCnet SHAZAM by including in your program the following command:

plot cquant kquant rquant pquant squant year / gnu line &
   commfile=plot.gnu datafile=plot.dat

After you exit the program, choose the GNUPLOT for Windows icon from the Econ 143 folder. Open the file you identified as your commfile= file (here, plot.gnu). If you wish, you can edit this file first, using TED. The file you actually edit will be called C000.gnu, which is what is loaded by your plot.gnu program. [A VERY good idea is to put your name in the title of the plot, so that you can find your own output as it emerges from the laser printer in the lab!]