Assignment 5 Economics 31 Fall 1999

Normal Curves, Z-tests, and Sample Distributions

Reading: Mirer Chpt. 9,10, pg.218-224, 18.1 Beals Chpt 6

1. Seniors at New Canaan High School who take the SAT's have scores that
are normally distributed with a mean of 1130 and a standard deviation of
200. Seniors at Greenwich High School, have scores that are normally
distributed with a mean of 1078 and a standard deviation of 250. A student
qualifies for a state honorary society if his or her score exceeds 1490.

a. For a randomly chosen senior from New Canaan, what is the probability
that his or her score on the test will qualify the student for the state
honorary society?

b. For a randomly selected senior from Greenwich, what is the probability
that his or her score will qualify the student for the state honorary society?

c. If we randomly and independently select from each high school, what is
the probability that at least one of these two students qualifies for the
state honorary society?

2.The percentage of children born in 1995 in Illinois to unwed mothers was
.34. If you sampled 100 births in 1995, what is the probability that
between 20 and 30 of them were born to unwed mothers?
(http://www.childrensdefense.org/states/data_il.html#population)

3. A computer firm offers a paid leave of absence to its engineers who wish
to get an advanced degree. However, applicants are tested and their
aptitude test scores must indicate a superb chance of success (as defined
by z-scores of +2 or better). In one test, applicants A through F earned
raw scores of 500, 631, 760, 438, 598, and 720. The mean score of all 200
applicants was 520, the standard deviation was 60. Who among the six will
go back to school?

4. The mean length of airplane flights from New York to Washington is 72
minutes with a standard deviation of 15 minutes (normal distribution).

a) What proportion of all flights last between 60 and 80 minutes?

b) Thirty three percent of all flights last longer than how many minutes?

c) What proportion of all flights last less than 45 minutes?

d) If 100 flights are chosen at random, what is the probability that at
least 25 of these flights last less than 60 minutes?

5. You speculate in foreign currencies. You hold a 30-day option to buy
pounds sterling at \$1.50 a pound and you have sold a 30-day contract to
sell pounds sterling at \$1.32. You will exercise your buy option if the
current price of the pound goes above \$1.50 because you will be able to
resell any pounds that you have purchased and pocket the difference between
the current price and \$1.50. Similarly, if the pound is above \$1.32, you
know that the holder of the contract you wrote to sell will exercise his
option. If the distribution of possible current prices for the pound was
approximately normal for the 30-day period with a mean of \$1.40 and a
standard deviation of 10 cents, what is the probability that you will have

6. Suppose the average yield of the stocks listed in the Standard and
Poor's index of 500 leading companies was 15% over the past year with a
standard deviation of 3%. The yield is made up of dividend, changes in the
price of the stock, and stock splits. Your investment advisor suggested a
portfolio of 10 stocks from the Standard and Poor's list at the beginning
of the year and the average yield of these stocks was 9%. If you had chosen
10 stocks at random from the list of 500, what is the probability that you