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Econ 57
Economic Statistics
Spring 2006
HW 4 assignment
1) a) What is the probability of getting 10 females in a sample of 50 people if p= % female=.40?
b) What is the probability of getting 15 females in a sample of 50 people if p= % female=.40?
c) What is the probability of getting 20 females in a sample of 50 people if p= % female=.40?
d) What is the probability of getting 25 females in a sample of 50 people if p= % female=.40?
e) What is the probability of getting 30 females in a sample of 50 people if p= % female=.40?
f) What is the probability of getting 18 to 22 females in a sample of 50 people if p=% female=.40?
Graph the results from above by hand using a bar graph.
g) Contrast the result from (c) to the probabilities of getting 40 females in a sample of 100 people and 400
females in a sample of 1000 people. What happens to the probability of getting exactly 40% female when
sample size is 100 and 1000? Explain.
i) What happens to the probability of getting between 36 to 44% females when n=100 and n=1000? Explain.
Graph these results.
j) Explain why this probability distribution looks bell-shaped (hint: think about the Central Limit Theorem).
k) What statistical concepts are illustrated by this exercise (besides the CLT)?
2) If the mean S&P500 stock return is 12% with a 20% standard deviation, then what is the probability of the
S&P500 returning over 22% in any given year? And what is the probability of the S&P 500 returning less than
5% in any given year? Draw a diagram illustrating the question and also show your work explicitly.
3) The admissions officer of Prestige U is trying to determine how many students to admit in order to achieve a
freshman class of 400 students. Using data on matriculation rates in matric.xls, calculate the optimal number of
admittees and explain your approach. Will your number guarantee a freshman class of exactly 400? What risks
do you run by admitting this many students to Prestige U?
4) The US Postal Service (USPS) has on-time standards for its mail delivery; for instance, letters from Washington
DC to Manhattan should arrive within 2 days. The USPS claims that it delivers 90% of the mail on time in
Manhattan. In the summer of 1988, New York magazine tested this claim by mailing 144 letters from around the
country to Manhattan zip codes and found that 50% of these letters did not arrive on time. If the USPS’s claims
are correct, what is the probability that 50% or more of these 144 letters will not arrive on time? Assume that the
binomial model applies and use a normal approximation.
5) How do samples of different sizes perform in terms of giving an accurate representation of the underlying
population? Does it depend on how large the sample is relative to the size of the population or does it depend on
the absolute size of the sample? Let’s see. First, examine a population of size 400 (N=400). Assume there are
40% female and 60% male in the population and assume the students numbered 1-160 are female and the students
numbered 161-400 are male. Second, examine a population of size 1,000 (N=1,000). Assume there are 40%
female and 60% male in the population and assume the students numbered 1-400 are female and the students
numbered 401-1,000 are male. Third, examine a population of size 10,000 (N=10,000). Assume there are 40%
female and 60% male in the population and assume the students numbered 1-4000 are female and the students
numbered 4001-10,000 are male. For these three situations, record the fraction of females (in the bold bordered
cells below) that you get for the samples of various sizes for the different sized populations. (HINT: use SSP.
First type in variable names, A-L, at the top of each of the first 12 columns (in place of var1, var2, var3, etc.)
Then choose: Uncertainty > random numbers > choose “Draw a random sample without replacement” > enter
population size in first box and sample size in second box and choose to put random sample in spreadsheet >
press OK after choosing the appropriate variable name to save the data under > press numerical order button at
bottom of screen. Then repeat process for other cells.)
sample
size, n
N=40
N=100
N=250
N=400
N=1000
N=2500
N=10000
N=400
n/N
N=400
% female
N=1000
n/N
N=1000
% female
N=10,000
n/N
N=10,000
% female
40/400 = 10%
100/400=25%
A)
B)
40/1000 = 4%
100/1000=10%
250/1000=25%
D)
E)
F)
40/10000 = .4%
100/10000 = 1%
H)
I)
400/400=100%
C)
1000/1000=100%
G)
1000/10000 = 10%
2500/10000 = 25%
10000/10000=100%
J)
K)
L)
Think about the following questions. Answer them the best you can anticipating what you will see in class. And
then modify your answers after you see the results for the whole class.
i) Compare your results to others in the class. For which cells are the results similar and for which cells are they
not? Explain.
ii) What do you notice as you compare the distributions of A to B to C or the distributions of E to F to G or the
distributions of J to K to L? Explain what is going on as you move from low n/N samples to 100% samples?
iii) What do you notice as you compare these 10% samples, A, E, and J?
iv) What do you notice as you compare these 25% samples, B, F, and K?
v) What do you notice as you compare these 3 distributions with sample size n=40: A, D, and H?
vi) What do you notice as you compare these 3 distributions with sample size n=100: B, E, and I?
vii) What happens to the concentration of the sampling distribution as n rises, holding n/N fixed (see iii and iv)?
viii) What happens to the concentration of the sampling distribution as n/N rises, holding n fixed (see v and vi)?
ix) In general, for the more typical situation where the population is very large and the sample is less than a
quarter of the population, does the accuracy of the point estimate depend on how big the sample is as a
fraction of the population or on the absolute size of the sample?
x) Is this statement true or false? Answer based on what you learned from the exercise above: “A 10% sample
should be sufficient in all situations. For a population of size 600, a survey with 76 people should be more
than enough to get a sample estimate that is representative of the population.”