Problem 1 - Properties of Estimators 1. A group of 7 roommates goes into a store to buy a closet....

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Problem 1 - Properties of Estimators 1. A group of 7 roommates goes into a store to buy a closet. They pick a closet but do not find any information on its exact (true) length in inches. To estimate its true length, each of them measures the closet with a tape line. Suppose every subject’s measurement Yi :“Length measure taken by person i [in inches]” ~ i.i.d N(µ, s2 ), where µ corresponds to the closet’s true length. Consider the following three estimators for µ: Tˆ 1 = Y2 Tˆ 2 = 1 n Xn i=1 Yi Tˆ 3 = 1 5 h 1 n - 1 nX-1 i=1 Yi i + 4 5 Yn (a) Which of these estimators are unbiased? (b) If possible, compare these estimators with respect to their relative
	
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Due on Monday, Oct. 2 5:00 p.m.
Fall 2017
Please round numbers to 4 decimal places. Write down your answers
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1 Problem 1 - Properties of Estimators
1. A group of 7 roommates goes into a store to buy a closet. They pick
a closet but do not ?nd any information on its exact (true) length in
inches. To estimate its true length, each of them measures the closet
with a tape line.
Suppose every subject’s measurement Y :“Length measure taken by
i
2
person i [in inches]” ~ i.i.d N(µ,s ), where µ corresponds to the
closet’s true length. Consider the following three estimators for µ:
ˆ
T =Y
1 2
n
X
1
ˆ
T = Y
2 i
n
i=1
n-1
h i
X
1 1 4
ˆ
T = Y + Y
3 i n
5 n- 1 5
i=1
(a) Which of these estimators are unbiased?
(b) If possible, compare these estimators with respect to their relative
e?ciency.
(c) Which of these estimators are consistent?
12. The company that produces tape lines wants to estimatep, the fraction
of defective tape lines in their ?nal delivery.
The random variable X : “Number of defective lines per examination
i
i” is assumed to follow a Bernoulli distribution with success probability
p.
(a) Write down E[X ] and V [X ] as a function of p.
i i
Using a simple random sample ofn examined tape lines (i.e. X ~
i
i.i.d., i = 1,...,n), the two following estimators are considered:
n
X
1
ˆ
T = X
1 i
n
i=1
1
ˆ
T = (X +X )
2 1 n
2
(b) Which of these estimators are unbiased?
ˆ
(c) Under which condition is T more e?cient?
1
(d) Which of these estimators are consistent?
2
3. Suppose X follows some distribution with E[X ] =µ and V [X ] =s .
i i i
P
1 n
2 2 2
¯
Show that S = (X -X) is an unbiased estimator for s .
i
i=1
n-1
Hints:
P P
n n
2
¯ ¯ ¯
• X =nX. Therefore, XX =nX .
i i
i=1 i=1
2 2 2 2
¯ ¯ ¯ ¯ ¯ ¯
• V (X) =E[X ]-E[X] , therefore E[X ] =V (X) +E[X] .
22 Problem 2 - Hypothesis Testing
Company ABC thinks about introducing a new environment-friendly light
bulb on the market with a reduced...

Robert · Accepted Answer

2) Given number of defective lines per examination i” is assumed to follow a Bernoulli 
distribution with success probability π therefore,  
a)  
E (Xi) = 0*(1- π) + 1* π = π 
E (Xi^2) = 0^2*(1- π) + 1^2* π = π 
V(X) = E (Xi^2) - [E (Xi)] ^2 = π - π^2 = π (1- π) 
b)
c)

Problem 1 - Properties of Estimators 1. A group of 7 roommates goes into a store to buy a closet. They pick a closet but do not find any information on its exact (true) length in inches. To estimate...

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