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)=s√n. 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