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Nov 5th Sampling Distribution of Sample Variance point and interval estimate-1

Nov 5th Sampling Distribution of Sample Variance point and interval estimate-1

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Sampling Distribution of Sample Variance Consider a random sample of n observations drawn from a population with unknown mean µ and unknown variance σ 2 , when x 1 , x 2 , …,x n are sample members 𝑠𝑠 2 = 1 ( 𝑛𝑛 − 1) ( 𝑥𝑥 𝑖𝑖 − 𝑥𝑥̅ ) 2 𝑛𝑛 𝑖𝑖=1 is called sample variance and the square root, s , is called sample standard deviation. Different random samples have different sample variance. The Expected value of the sample variance is the population variance 𝐸𝐸 [ 𝑠𝑠 2 ] = 𝜎𝜎 2 If we assume the underlying population distribution is normal, then it can be shown that the sample variance and the population variance are related through a probability distribution known as chi-square distribution

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Chi-square distribution of sample and population variance using random sample of size n observations from a normally distributed population whose variance is 𝜎𝜎 2 and resulting sampling variance is 𝑠𝑠 2 , it can be shown that 𝛸𝛸 ( 𝑛𝑛−1 ) 2 = ( 𝑛𝑛 − 1) 𝑠𝑠 2 𝜎𝜎 2 = ( 𝑥𝑥 𝑖𝑖 − 𝑥𝑥̅ ) 2 𝜎𝜎 2 had a distribution known as chi-square ( 𝛸𝛸 ( 𝑛𝑛−1 ) 2 ) with (n – 1) degrees of freedom. The distribution is asymmetric and defined for only positive values since variance are all positive values. We can characterize a particular member of the family of the chi-square distributions by a single parameter referred to as degree of freedom, denoted as v . A chi-square distribution with v degrees of freedom is denoted 𝛸𝛸 𝑣𝑣 2 . The mean and variance of this distribution are equal to the number of degrees of freedom and twice the number of degrees of freedom E [ 𝛸𝛸 𝑣𝑣 2 ] = 𝑣𝑣 and Var [ 𝛸𝛸 𝑣𝑣 2 ] = 2 𝑣𝑣

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