MBACATÓLICA
QUANTITATIVE METHODS FOR BUSINESS
September-December 2006
Assignment #2
Problem 1
A common fear of many people is that they will be struck by lightning. Portugal has about 10,5
million people and last year, a normal year, there were an estimated 5 people struck. Based on
that data, and assuming such events are independent from one another, what are the
probabilities that
a) Zero persons will be struck by lightning next year?
b) Less than 8 people will be struck by lightning?
c) 10 or more people will be struck by lightning?
Assume that the probability of one person being struck is independent from another person
being struck. (Hint: Use an approximation to the correct probability distribution)
Problem 2
Economists have been studying the birth and death rates for firms. In Portugal, 1998 (a
representative year), the overall percentage of new firms that did not last for more than a year
was about 20.5%. Suppose you are the Bank MA regional Director responsible for the “loans to
small firms” and you have 300 new firms applying for loans.
a) What is the median number of new firms that you expect will still be around
next year? What is the probability that 250 or more firms will survive for more
than a year?
b) Now assume that you ended up lending to 40 new firms in the restaurant
business, and that 12 were officially closed by the end of the year. Provide a
95% confidence interval for the “one year survival rate” in the restaurant
business.
c) Test if firms in the restaurant business have the same “one year closing rate” as
firms in the general economy.
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Problem 3
“The sample size is a decision variable reflecting the conflict between
inference precision/error and the cost of sampling.”
To illustrate this statement, consider that credit demand to Bank XXX is approximately
normally distributed with standard deviation equal to 15.
a) In constructing a 95% confidence interval for the mean credit demand, what should be
the minimum sample size to guarantee a maximum error margin of 3?
b) Determine the minimum sample size to guarantee that the probability of type II
error is not larger than =10%, when testing H0: =10 vs. HA: =13 (=5%).
And if =5%?
Consider now that Bank XXX classifies localities according to the probability of default of their
clients: that probability can be either 5% or 7.5%. The bank is opening a branch in a new
locality and wants to infer the probability of default of the new clients.
c) Determine the minimum sample size to guarantee that the probability of type II
error (incorrectly not rejecting that the sample is from a “5% locality”), is not
larger than =3% (=1%).
Problem 4
Corruption is believed to depend both on the level of inflation and on the population standard of
living. An annual Corruption Perceptions Index (CPI) has been released since 1995 and ranks
countries by their perceived levels of corruption, as determined by expert assessments and
opinion surveys. The CPI Score and ranges between 10 (highly clean) and 0 (highly corrupt).
The file HW2_data.xls contains the following series for 132 countries:
CPI: the CPI Score for 2004
GDPpc: 2004 gross domestic product per capita based on purchasing-power-parity (1000$)
INFL: average inflation rate for the period 2000-04
a) Regress the CPI Score on:
(i)
GDPpc
(ii)
INFL
(iii) GDPpc and INFL
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Which of the three models you prefer? Justify your answer and comment the
sign of the estimates.
b) Regress ln(CPI) on ln(GDPpc) and ln(INFL):
(i)
Interpret the meaning of the slope coefficients.
(ii)
Test (=5%) if the coefficients are significant.
(iii) Test (=5%) if the coefficient associated with ln(GDPpc) can be 0.3.
c) Perform two additional regressions to explain ln(CPI) by ln(GDPpc) and ln(INFL),
(i)
one allowing for regional differences in the constant and
(ii)
the other allowing for regional differences on the impact of the
standard of living on the corruption index,
classifying countries as “developed” (Europe, North America and Oceania) and
“others”. Which model represents better this regional factor? Justify your answer.
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