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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 = [(X1 - Y)^2 + ... + (Xn - Y)^2] / (n - 1)

where Y is the mean of the observations {X1, ..., Xn}, 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 chi-square distribution with n - 1 degrees of freedom.

To test the hypotheses

H0: ø^2 = þ

against

H1: ø^2 != þ

determine where on the chi-square density the computed value of Z falls. With a sample of 16 observations, we can reject H0, 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 F-distribution with (n - 1) and (m - 1) degrees of freedom.

If n = 20 and m = 10, then the F19,9 distribution is:

Reject the equality of the variances, at the 90% level, if the F statistic lies to the right of 2.42.

 
   

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