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Question 2 part c in the attachment, since i cant copy paste it.


Now, consider the following two potential estimators for the variance, σ2, of a Normal population,each based on the sample of n (> 2) observations, obtained by simple random sampling:    nii x xns12 2 ( ) ( 1)1 ; and  nii xn 12 2 ( ) 1ˆ(c) We proved in class that s2 is an unbiased estimator of σ2. Using part (a) above, prove that2 ˆ is a biased estimator of σ2. What is the expression for its bias? What is the sign of this bias? Can the bias ever be zero? (Explain.)

ECON 246 University of Victoria
Statistical Inference (Section A01)
Assignment 2, Spring 2017 Due: Thursday 2 February, at 4:00p.m.; in the box for your lab. section outside the
Department of Economics office. Question 1:
(a)
For a random variable, C, which has chi-square distribution with 8 degrees of freedom,
use the table of chi-square values on CourseSpaces, or from your textbook to
determine the values of a and b such tha Pr .[a C b] 0.90 , and such that the areas
in each tail are equal.
(5 marks)
(b)
Suppose that we take a simple random sample of size n = 6 from a population that is
Normal with a mean of μ = $38.75, and a standard deviation of σ = $3. Using EViews,
1 n
2
compute the probability that the sample variance (s2 = ( xi x ) ) will lie between
n 1 i 1
$2 8.0 and $2 11.0 .
(5 marks)
Question 2:
(a)
Let Y be any random variable, whose mean is E[Y] = μ, and whose variance is σ2. Show
that σ2 = E(Y2) – μ2.
(2 marks) (b) Show that E[(Y – μ)4] = E(Y4) - 4μE(Y3) + 6μ2σ2 + 3μ4 .
(6 marks) Now, consider the following two potential estimators for the variance, σ2, of a Normal population,
each based on the sample of n (> 2) observations, obtained by simple random sampling: s2 (c) 1 n
2 ( xi x )
( n 1) i 1 ; and ˆ 2 1 n 2 ( xi )
n i 1 We proved in class that s2 is an unbiased estimator of σ2. Using part (a) above, prove that
ˆ 2 is a biased estimator of σ2. What is the expression for its bias? What is the sign of
this bias? Can the bias ever be zero? (Explain.) (6 marks)
Question 3:
For this question you’ll need to use the following EViews workfile and EViews program file on
the server:
S:Social SciencesEconomicsEcon246ass2.wf1
S:Social SciencesEconomicsEcon246ass2.prg
(These files are also on CourseSpaces.
First, open the workfile in the usual way. Then select File / Open / Programs… and open the
above program file. The program will open in a new window, separate from your workfile, but 1 the two files will “communicate” with each other. In particular, when you press the “Run” tab in
the program file, the results will appear in the Workfile.
(a) Edit the first line in the program, as instructed in the program file itself. That is, alter
this line so that instead of rndseed 123456 it becomes rndseed ??????, where ??????
is the last 6 digits of your student number. (This will ensure that everyone has a different
“seed” for the random number generator, and therefore has different data and results.)
Now Run the program. You’ll see a window like this. Select the “Quiet (fast)” mode: Report the results (the histograms & summary statistics) for the sampling distributions of
the sample mean and the other sample statistic that you have created. What is this other
statistic?
(3 marks)
Note that the population data for X are random values created by taking the sum of
two (independent) random variables. In the population, what are the values of the
mean, μ, and the variance, σ2, of X ?
(3 marks)
(c)
What values would you expect to see for the mean and the variance of the sampling
distribution of x ? Compare these values with the mean and variance of your “simulated”
sampling distribution for x .
(3 marks)
(d)
Now increase the sample size to n = 100, and report the two sampling distributions.
(3 marks)
(e)
What values would you expect for the mean and the variance of the sampling distribution
of x now? Compare these values with the mean and variance of your “simulated”
(2 marks)
sampling distribution for x .
(4 marks)
(f)
Repeat parts (d) and (e) with a sample of size n = 1,000.
Very briefly, explain how your results illustrate the “Central Limit Theorem”.
(g)
(3 marks)
(h)
On the basis of your results, do you think that the second sample statistic whose sampling
distribution you’ve created is an unbiased estimator of the mean, μ, of this particular
population? Do think it is a consistent estimator of μ? (Explain.)
(3 marks)
(i)
Which of the two estimators is relatively more efficient, when the sample size is 1,000?
(2 marks)
Note: Each student will get different answers for the various parts of this question.
Total: 50 marks
(b) 2

 


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