Statistics Assignement 5 1

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					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
     this works.

      (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.


Description: Prof Rushen's Notes for MBA/ BBA students