Unbiased estimation of μ by xˉ and of σ2
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!
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
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
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:
Exercises checked off
Unbiased estimation of μ by xˉ and of σ2
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!
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
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
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:
Exercises checked off