Econ 221 - Introduction to probability and statistics I - Sections 2/5
Assignment 5, due Friday December 28th, before class starts.
Solve the following problems from ”Statistics for business and economics”, 6th edition.
• 7.65, 7.67, 7.75
• 8.13, 8.25, 8.47, 8.49, 8.57, 8.59
Problem using MsExcel.
Example 8.1 compares theoretically two unbiased estimators of the population mean
(the sample mean and the sample median) when the population is normally dis-
tributed. You will ﬁrst verify this result using Monte Carlo simulations. (This method
is used in your textbook (pp. 244-248) to illustrate the Central Limit Theorem.) Then
you will analyze what happens if the the population is not normally distributed. You
will ﬁnally approximate conﬁdence intervals for the median.
1. The book claims that the relative eﬃciency of the mean with respect to the
median is 1.57. To verify that claim, you need to approximate the sampling
variance of each estimator and consider the ratio. The quality of the approx-
imation you will make in this problem depend on the number of samples you
consider. Here, I ﬁxed that number to 2000, which should be enough to see how
(a) Use MsExcel to generate 2000 samples of 25 observations from a standard
normal population (/tools/data analysis/random number generation).
(This will take some time, be patient!) For each sample, compute the
sample mean (=average()) and median (=median()). You then have
2000 samples means and medians. Then compute the sampling variances
(=var()) of the sample mean and median. Is the ratio close to 1.57?
(b) For the sample median to be an unbiased estimator of the population mean,
the population distribution must be symmetric. Repeat the exercise above
with samples from a uniform population on the [0,1] interval. Is the sample
mean or the sample median more eﬃcient in this case? What is the relative
(c) Repeat the same exercise again for samples from a fair Bernoulli popu-
lation. Is the sample mean or the sample median more eﬃcient in this
2. You will now use MsExcel to approximate conﬁdence intervals. You will ﬁrst
approximate an interval for a case where you know the true interval to make
sure you understand the method.
(a) Put the 2000 sample means you obtained in part a) in increasing order
(/data/sort). Note that 90% of your sample means are between the
100th and the 1900th sample mean (ordered). Use these values as an
approximate 90% conﬁdence interval. Compare that interval to the exact
interval obtained from the normal probability table.
(b) You can use that trick to approximate the sampling distribution of ANY
statistics for ANY population distribution. Approximate a 90% conﬁdence
interval for the sample median with a sample size a 25 when the population
uniform on the [0,1] interval..
Hand in one hard copy per team of 2-3 students.