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safina
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Homework Statement
A sample of size n is drawn from a population having N units by simple random sampling without replacement. A sub-sample of size [tex]n_{1}[/tex] units is drawn from the n units by simple random sampling without replacement. Let [tex]\bar{y_{1}}[/tex] denote the mean based on [tex]n_{1}[/tex] units and [tex]\bar{y_{2}}[/tex] based on (n-[tex]n_{1}[/tex]) units.
Consider the estimator [tex]\hat{\overline{Y}}[/tex] = w[tex]\bar{y_{1}}[/tex] + (1-w)[tex]\bar{y_{2}}[/tex].
Show that E[[tex]\hat{\overline{Y}}[/tex]] =[tex]\overline{Y}[/tex] and obtain its variance.
Homework Equations
The Attempt at a Solution
E[[tex]\hat{\overline{Y}}[/tex]] = E[w[tex]\bar{y_{1}}[/tex] + (1-w)[tex]\bar{y_{2}}[/tex]]
= w E[[tex]\bar{y_{1}}[/tex]] + (1-w) E[[tex]\bar{y_{2}}[/tex]]
= w[tex]\overline{Y}_{1}[/tex] + (1-w)[tex]\overline{Y}_{2}[/tex]
Why I did not arrive at the rigth answer which is [tex]\overline{Y}[/tex]?