chap06S4PRN

# 1 s e 2 x that is the divisor of n 1 ensures that 2 x

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1 s E 2 X That is, the divisor of (n – 1) ensures that 2 X s gives an unbiased estimation rule for the population parameter 2 σ . This justifies the use of (n – 1) in the divisor.

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Econ 325 – Chapter 6.4 3 Another statistical result about probability distributions is needed. Let 1 Z , 2 Z , . . . , m Z be a set of independent standard normal random variables. Define the random variable: = = m 1 i 2 i Z C C has a 2 χ (chi-square – pronounced ki-square) distribution with m degrees of freedom. Properties of the probability density function for the chi-square distribution are: defined only for values 0 , a skewed shape that depends on the degrees of freedom. Econ 325 – Chapter 6.4 4 PDF for the 2 χ distribution with 4, 6 and 8 degrees of freedom (df). 0 2 4 6 8 10 12 14 4 df 6 df 8 df Critical values or cut-off points of the chi-square distribution are listed in a textbook Appendix Table.
Econ 325 – Chapter 6.4 5 Assume that 1 X , 2 X , . . . , n X are normally distributed random variables. Define the random variable: 2 = 2 σ - = σ - n 1 i 2 i 2 X ) X X ( s ) 1 n ( This random variable has a chi-square distribution with (n – 1) degrees of freedom. Probability statements about the variance can now be made.

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1 s E 2 X That is the divisor of n 1 ensures that 2 X s...

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