22.2 Sampling distribution: One mean with population standard deviation unknown

When a sample mean is used to estimate a population mean, the sample mean will vary from sample to sample: sampling variation exists, as we saw in the previous section.

When we do not know the population standard deviation σ (which is almost always the case), we estimate it using the sample standard deviation s. Then, the standard error of the sample mean is s.e.(ˉx)=sn. With this information, we can describe the sampling distribution of the sample mean.

Definition 22.1 (Sampling distribution of a sample mean) When the population standard deviation is unknown, the sampling distribution of the sample mean is described by:

  • an approximate a normal distribution,
  • centred around μ,
  • with a standard deviation (called the standard error of the mean) of
s.e.(ˉx)=sn, when certain conditions are met, where n is the size of the sample, and s is the standard deviation of the individual observations in the sample (that is, the sample standard deviation).