Further Hypothesis Tests

This chapter is about testing spread, not average. Earlier in your stats course you tested whether a mean had changed. Here, you test whether a variance has changed — either on its own, or by comparing two variances against each other.

There are really only three tools in this chapter, and they build on each other in a straight line:

• The chi-squared (χ²) distribution — describes how the sample variance S² of a single normal sample behaves. Used for confidence intervals and hypothesis tests on one variance.
• The F-distribution — describes what happens when you take the ratio of two sample variances. It has two "degrees of freedom" instead of one, because two samples are involved.
• The F-test — the actual hypothesis test that uses the F-distribution to decide whether two populations really do have equal variance.

Why we need a new distribution

You already know that the sample mean X̄ is an unbiased estimator of the population mean μ. In exactly the same spirit, the sample variance

Different samples from the same population give different values of s² — it's a particular observed value of the random variable S². To build a confidence interval for the true population variance σ², we need to know how S² is distributed. That's where chi-squared comes in.

Building the confidence interval

Because (n−1)S²/σ² follows a χ² distribution, we can trap it between two critical values from the χ² table with a chosen probability. Rearranging for σ² flips which critical value goes on top and bottom — this trips people up constantly, so watch closely in the example.