28.3 About sampling distributions and expectations

The sampling distribution describes, approximately, how the sample statistic (such as \(\bar{x}\)) is likely to vary from sample to sample over many repeated samples, when \(H_0\) is true: it describes the sampling variation. Under certain circumstances, sampling distributions often have an approximate normal distribution, which is the basis for computing \(P\)-values (or approximating \(P\)-values using the 68–95–99.7 rule).

When the sampling distribution is described by a normal distribution, the mean of the normal distribution is the parameter value given in the assumption (\(H_0\)), and the standard deviation of the normal distribution is called the standard error.

In some cases, the sample statistic may not have a normal distribution, but a quantity easily derived from the sample statistic does have a normal distribution (for example, the odds ratio11).

  1. In this case, the logarithm of the odds ratio has an approximate normal distribution.↩︎