ST3239: Survey Methodology - The Department of Statistics and ...

Proof.
Es 2 =
=
=
(
1 n
)
∑
Eyi 2 − nE(ȳ) 2
n − 1
i=1
(
1 ∑ n [
V ar(yi ) + (Ey i ) 2] − n [ V ar(ȳ) + (Eȳ) 2])
n − 1
i=1
(
1
n [ [ ])
σ 2 + µ 2] σ
2
N − n
− n
n − 1
n N − 1 + µ2
= nσ2
n − 1
= Nσ2
N − 1
[
1 − 1 n
( N − n
N − 1
)]
= nσ2
n − 1
( )
nN − n − (N − n)
n(N − 1)
The next theorem is an easy consequence of the last theorem.
Theorem 2.2.4 ˆσ 2 := N−1
N
s2 is an unbiased estimator of σ 2 , e.g.
E
( N − 1
N s2 )
= σ 2 .
We shall define
f = n N
to be the sample proportion,
1 − f = 1 − n N
to be the finite population correction (ab. fpc)
Then we have the following theorem.
Theorem 2.2.5 An unbiased estimator for V ar(ȳ) is
Proof.
E ̂V ar(ȳ) = Es2
n
̂V ar(ȳ) = s2
n
(1 − f) .
(1 − f) =
Nσ2
n(N − 1) (1 − n N )
Confidence intervals for µ
It can be shown that the sample average ȳ under the simple random sampling is approximately
normally distributed provided n is large (≥ 30, say) and f = n/N is not too close to 0 or 1.
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