Statistical Theory: Estimating the Population Mean Wage in the Ann Arbor Area
A statistical assignment estimating population mean wages using statistical theory.
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Statistical Theory: Estimating the Population Mean Wage in the Ann Arbor Area
Part II: Statistical theory (12 points in total)
You would like to know the average wage of all working women and men between the ages of
18 and 54 in the Ann Arbor area. You find it too costly in terms of time and money to ask
everyone about their earnings, so you randomly select 500 people from this group, and ask them
about their wage.
(i) What are the population and the population parameter of interest in this example? What is the
sample, and what the sample size, N?
The population in this case is all working women and men between the ages of 18 and 54 in the
Ann Arbor area and the population parameter of interest in this example is the average wage of
them.
The sample is the randomly selected 500 people from the population and the sample size N =
500.
Now, comment on each of the proposed estimators of the population mean wage, μ :
(ii) using the sample mean as an estimator of the population mean
Is it unbiased? Show formally.
Note that, 𝐸(𝑋ത) = 𝐸 ቀଵ
ே ∑ 𝑋ቁ = ଵ
ே ∑ 𝐸(𝑋) = ଵ
ே ∑ μ = μ
Thus it is unbiased.
Is it efficient? Why or why not? Show that its variance equals ఙ2
ே
𝑉(𝑋ത) = 𝑉 ቀଵ
ே ∑ 𝑋ቁ = ଵ
ே2 ∑ 𝑉(𝑋) = ேఙ2
ே2 = ఙ2
ே ; Due to independence 𝑉(∑ 𝑋) = ∑ 𝑉(𝑋)
Thus it is an efficient estimator.
(iii) another estimator using only three observations from your sample and calculated
as: ଵ
ଷ (𝑋ଵ + 𝑋2 + 𝑋ଷ)
Is it unbiased? Show formally.
Part II: Statistical theory (12 points in total)
You would like to know the average wage of all working women and men between the ages of
18 and 54 in the Ann Arbor area. You find it too costly in terms of time and money to ask
everyone about their earnings, so you randomly select 500 people from this group, and ask them
about their wage.
(i) What are the population and the population parameter of interest in this example? What is the
sample, and what the sample size, N?
The population in this case is all working women and men between the ages of 18 and 54 in the
Ann Arbor area and the population parameter of interest in this example is the average wage of
them.
The sample is the randomly selected 500 people from the population and the sample size N =
500.
Now, comment on each of the proposed estimators of the population mean wage, μ :
(ii) using the sample mean as an estimator of the population mean
Is it unbiased? Show formally.
Note that, 𝐸(𝑋ത) = 𝐸 ቀଵ
ே ∑ 𝑋ቁ = ଵ
ே ∑ 𝐸(𝑋) = ଵ
ே ∑ μ = μ
Thus it is unbiased.
Is it efficient? Why or why not? Show that its variance equals ఙ2
ே
𝑉(𝑋ത) = 𝑉 ቀଵ
ே ∑ 𝑋ቁ = ଵ
ே2 ∑ 𝑉(𝑋) = ேఙ2
ே2 = ఙ2
ே ; Due to independence 𝑉(∑ 𝑋) = ∑ 𝑉(𝑋)
Thus it is an efficient estimator.
(iii) another estimator using only three observations from your sample and calculated
as: ଵ
ଷ (𝑋ଵ + 𝑋2 + 𝑋ଷ)
Is it unbiased? Show formally.
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Subject
Statistics