Further χ² tests & unbiased estimators
Unbiased estimation of μ by xˉ and of σ2 by Sx2, and chi-squared tests with grouped categories so all expected frequencies exceed 5 and degrees of freedom are reduced by 1 for each estimated parameter, using df=n−1−k.
Grouping data for Chi squared (χ²) tests
Before performing a χ2 test, it's important to verify that all expected frequencies are larger than 5. If any are not, categories must be combined before performing the test. For example:
Note that when we combine categories, the degrees of freedom decrease!
x̅ as an unbiased estimate of μ
If the true mean of some distribution is unknown, we can average samples taken from the distribution to produce an unbiased estimate of the population mean:
We call the estimate unbiased since
sₙ₋₁² as an unbiased estimator of σ²
If the true variance of some distribution is unknown, we can use the sample standard deviation to get an unbiased estimate of the population variance:
(Your calculator returns both Sx - which is the same as sn−1 and σx).
We call the estimate unbiased since
Chi squared (χ²) with estimated parameters
When we perform a χ2 goodness of fit test with unbiased estimates as parameters for some distribution, each estimated parameter is an additional constraint on the data, so we need to subtract from the degrees of freedom:
where k is the number of parameters estimated.
Nice work completing Further χ² tests & unbiased estimators, here's a quick recap of what we covered:
Skills covered
Exercises checked off