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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 AreaPart II: Statistical theory (12 points in total)You would like to know the average wage of all working women and men between theages of18 and 54 in the Ann Arbor area. Youfind it too costly in terms of time andmoney to askeveryone about their earnings, so you randomly select 500 people from thisgroup, and ask themabout their wage.(i) What are the population and the population parameter of interest in thisexample? What is thesample, 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 theAnn Arbor area and the population parameter of interest in this example is the average wage ofthem.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 meanas an estimator of the population meanIs it unbiased? Show formally.Note that,𝐸(𝑋ത)=𝐸ቀଵே∑𝑋௜ቁ=ଵே∑𝐸(𝑋௜)=ଵே∑μ=μThus it is unbiased.Is it efficient? Why or why not? Show that its variance equalsఙ2ே𝑉(𝑋ത)=𝑉ቀଵே∑𝑋௜ቁ=ଵே2∑𝑉(𝑋௜)=ேఙ2ே2=ఙ2ே; Due to independence𝑉(∑𝑋௜)=∑𝑉(𝑋௜)Thus it is an efficient estimator.(iii) another estimator using only three observations from your sample and calculatedas:ଵଷ(𝑋ଵ+𝑋2+𝑋ଷ)Is it unbiased? Show formally.

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Statistics

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