Final (make-up)
ECO 523
Statistics (Monsoon 2020)
Instructions: Read the following instructions carefully. This exam contains 4 questions, 3
pages for a total of 50 marks. You should be logged into MS Teams course page all times and
your camera should be on till the end. Write your answers on paper using a pen in legible
handwriting. At the top of the first page of your answer script, mention your name, roll
number and whether you are a masters or phd student. After finishing, scan your answers in
a single pdf file, name the file “yourname ECO523 final” and upload it to Blackboard. You
have 115 minutes to finish the exam. You should stop writing after that and you will get
an additional 15 minutes to upload your answers to Blackboard. After you have uploaded
the answers to Blackboard, you should send a one word message in the chat window of MS
Teams ”submitted”. Only after this, you can switch off your camera and log off from MS
Teams. Failure to abide by any of the above instructions will be penalized. You may use a
calculator. However, you do not need to simplify complicated expressions. Make use of the
Appendix if necessary!
1. Assume the following pdfs:
(
f (x) =
3 2
x
8
0
0≤x≤2
otherwise
fY |X (y|x) ∼ U nif (−1, X)
(a) (5 marks) Find the variance of X.
(b) (5 marks) Find E(Y)
2. You have a sample of size n, X1 , ...., Xn from a uniform distribution, U [θl , θu ] distribution.
(a) (5 marks) Assume that it is known that θl = 0. Find the Method of Moments
(MM) estimators of θu .
(b) (5 marks) Now assume that both θl and θu are unknown. Find the Method of
Moments (MM) estimators of both θl and θu .
(c) (5 marks) Determine whether the estimator in part (a) is unbiased and consistent.
3. (10 marks) Suppose that you are going to draw a random sample from a Bernouli distribution in an effort to estimate the parameter p. How large of a sample will you need to
draw in order to ensure that the width of the 95% confidence interval for your estimate
no greater than 0.01? (Hint: assume that n will be large and consider bounds on the
quantity p(1 − p).)
4. Suppose size of watermelons is normally distributed with a mean of 7 pounds and a
variance of 1 pound. A farmer is worried that the watermelons he is cultivating are
underweight i.e. the watermelons that he is cultivating are 1 pound lighter on average
(but still have the same variance) and calls you to investigate.You observe the weight
of a random sample of n = 10 watermelons from his field. The mean weight of the 10
watermelons is 6.2 pounds.
(a) (3 marks) Suppose you want to do a test of the null hypothesis that the watermelons grown by the farmer are not underweight against the alternative hypothesis
that they are. Write down the null hypothesis and alternative hypothesis mathematically.
(b) (4 marks) Perform a 5% test of the hypotheses from part (a).
(c) (4 marks) What is the power of the test in part (b)?
(d) (4 marks) Suppose you only knew the mean of the distribution of the weight, not
the variance, but you did have an estimate of the variance, S 2 = 1.5. Perform a 5%
test of the null against alternative.
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Appendix
1. Binomial distribution: X ∼ Bin(n, p)
P (X = k) = n Ck pk (1 − p)n−k
for k = 0, 1, . . . ,n (and P(X = k) = 0 otherwise).
2. Uniform distribution: U ∼ U nif (a, b)
(
f (x) =
1
b−a
0
a<x<b
otherwise
3. Normal distribution: X ∼ N (µ, σ 2 )
(x−µ)2
1
(exp)(− 2σ2 )
f (x) = √
2πσ
X
X
XX
4. V ar(
Xi ) =
V ar(Xi ) +
Cov(Xi , Xj )
i
i
i
i6=j
5. z0.9 = 1.28; z0.93 = 1.49; z0.95 = 1.65; z0.975 = 1.96; z0.99 = 2.33 where Φ(zα ) = α
6. t9 (0.9) = 1.38; t9 (0.95) = 1.83; t9 (0.99) = 2.82 where tn (p) is the pth percentile of a
t-distribution with n degrees of freedom.
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