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 + ... + (X_n - Y)^²] / (n - 1)

where Y is the mean of the observations {X₁, ..., 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 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 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.