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

to honor your sell contract but not exercise your buy option?

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

would have done as badly as or worse than your investment advisor?
Assume

that the distribution of yields is approximately normal.