Egwald Statistics: Probability and Stochastic Processes
by
Elmer G. Wiens
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Distributions  Sampling Distributions  Sample Mean  Hypotheses Testing  Random Walk
 References
Hypotheses Testing
Testing the mean  Testing the variance
B. Testing the Variance
The random variable S^^{2} defined by
S^^{2} = [(X_{1}  Y)^^{2} + ... + (X_{n}  Y)^^{2}] / (n  1)
where Y is the mean of the observations {X_{1}, ..., X_{n}}, is called the sample variance. S is called the standard error of the mean.
1. Sample from a normal population
The r.v. Z = (n  1) * S^^{2} / ø^^{2} has the chisquare distribution with n  1 degrees of freedom.
To test the hypotheses
H_{0}: ø^^{2} = þ
against
H_{1}: ø^^{2} != þ
determine where on the chisquare density the computed value of Z falls. With a sample of 16 observations, we can reject H_{0}, at the 95% level of confidence, if the Z statistic falls in either of the two critical regions: to the right of 27.488, or to the left of 6.262.
2. Comparing the variances of two populations
Suppose we compute sample variances S^^{2} and R^^{2} from samples of size n and m drawn from two populations. Assume that S^^{2 } >= R^^{2}. Then the statistic
F = S^^{2} / R^^{2}
has the Fdistribution with (n  1) and (m  1) degrees of freedom.
If n = 20 and m = 10, then the F_{19,9} distribution is:
Reject the equality of the variances, at the 90% level, if the F statistic lies to the right of 2.42.
