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Further χ² tests & unbiased estimators
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Using chi squared tests where numerical categories need to be combined, and goodness of fit tests using estimated parameters.
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Grouping data for Chi squared (χ²) tests
AHL AI 4.12
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
AHL AI 4.14
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:
μ≈xˉ=n∑xi.
We call the estimate unbiased since
μ=xˉ as n→∞
sₙ₋₁² as an unbiased estimator of σ²
AHL AI 4.14
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:
σ2≈Sx2=sn−12=n−1nσx2
(Your calculator returns both Sx - which is the same as sn−1 and σx).
We call the estimate unbiased since
σ2=Sx2 as n→∞
Chi squared (χ²) with estimated parameters
AHL AI 4.12
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:
df =n−1−k
where k is the number of parameters estimated.
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