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Sampling Distribution of Sample Variance

Like sample mean, if we calculate the sample variance for each samples drawn from a population then it shows also a random variable. We have the following result: If a random sample of size n with sample variance S2 is taken from a normal population having the variance σ 2, then

is a random variable having the chi-square distribution with the degrees of freedom v = n - 1.

(In chi-square distribution table Xarepresents the area under the chi-square distribution to its right is equal to a).

If S12 and S22 are the variances of independent random sample of size n1 and n2 respectively, taken  from two normal populations having the same variance, then

F = S12 / S22

is a random variable having the F distribution with the degrees of freedoms v1 = n1 - 1 and v2  = -1.

Example 4. If two independent random samples of size n1 = 9 and n2 = 16 are taken from the normal population, what is the probability that the variance of the first sample will be at least four times as large as that of the second sample ?

Solution. Here v 1 = 9 - 1 = 8, v2 16- 1 = 15

S12  = 4 S22

From F distribution table we find that

F0.01 = 4.00 for                         v1= 8 and v2 = 15

Thus, the desired probability is 0.01.