Which of the following statements are true if $X _ { 1 } , X _ { 2 } , \cdots \cdots , X _ { n }$ is a random sample from a distribution with mean $\mu \text { and variance } \sigma ^ { 2 }$ ?
A) $S ^ { 2 } = \sum ( X - \bar { X } ) ^ { 2 } / ( n + 2 ) \text { is an unbiased estimator of } \sigma ^ { 2 }$
B) $S ^ { 2 } = \sum ( X - \bar { X } ) ^ { 2 } / ( n + 1 ) \text { is an unbiased estimator of } \sigma ^ { 2 }$
C) $S ^ { 2 } = \sum ( X - \bar { X } ) ^ { 2 } / n \text { is an unbiased estimator of } \sigma ^ { 2 }$
D) $S ^ { 2 } = \sum ( X - \bar { X } ) ^ { 2 } / ( n - 1 ) \text { is an unbiased estimator of } σ^ { 2 }$

E) All of the above statements are true provided that the sample size n 30.

Question

Which of the following statements are true if $X _ { 1 } , X _ { 2 } , \cdots \cdots , X _ { n }$ is a random sample from a distribution with mean $\mu \text { and variance } \sigma ^ { 2 }$ ?
A) $S ^ { 2 } = \sum ( X - \bar { X } ) ^ { 2 } / ( n + 2 ) \text { is an unbiased estimator of } \sigma ^ { 2 }$
B) $S ^ { 2 } = \sum ( X - \bar { X } ) ^ { 2 } / ( n + 1 ) \text { is an unbiased estimator of } \sigma ^ { 2 }$
C) $S ^ { 2 } = \sum ( X - \bar { X } ) ^ { 2 } / n \text { is an unbiased estimator of } \sigma ^ { 2 }$
D) $S ^ { 2 } = \sum ( X - \bar { X } ) ^ { 2 } / ( n - 1 ) \text { is an unbiased estimator of } σ^ { 2 }$

E) All of the above statements are true provided that the sample size n > 30.

Quizplus · Accepted Answer

The correct formula for an unbiased estimator of the variance $\sigma^2$ in a sample is $S^2 = \sum (X - \bar{X})^2 / (n - 1)$, where $n$ is the sample size. This adjustment with $n - 1$ in the denominator is known as Bessel's correction, which corrects the bias in the estimation of the population variance from a sample.

Which of the Following Statements Are True If X1,X2,⋯⋯ ,XnX _ { 1 } , X _ { 2 } , \cdots \cdots , X _ { n }X1​,X2​,⋯⋯,Xn​

Which of the Following Statements Are True If $X _ { 1 } , X _ { 2 } , \cdots \cdots , X _ { n }$