TUTORIAL 2 SOLUTIONS #7.7.11 Consider a population of size four ...
TUTORIAL 2 SOLUTIONS #7.7.11 Consider a population of size four ...
TUTORIAL 2 SOLUTIONS #7.7.11 Consider a population of size four ...
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c. From a. we have<br />
E(n −1 s 2 )= 1 n E(s2 )<br />
= σ2<br />
n<br />
= σ 2¯X .<br />
That is n −1 s 2 is an unbiased estimate <strong>of</strong> the<br />
variance <strong>of</strong> ¯X.<br />
d. Recall that T = N ¯X and Var(T )is<br />
given by<br />
Consequently,<br />
σ 2 T<br />
=Var(N ¯X)<br />
= N 2 Var( ¯X)<br />
= N 2σ2<br />
n .<br />
E(n −1 N 2 s 2 )=n −1 N 2 E(s 2 )<br />
= n −1 N 2 σ 2<br />
= σ 2 T .<br />
6
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