28.4 About observations and the test statistic
The sampling distribution describes what values the sample statistic can reasonably be expected to have, over many repeated samples. Since the sampling distribution of the statistic has an approximate normal distribution under certain conditions, the observed value of the sample statistic can be expressed as a something like a \(z\)-score (called a \(t\)-score when the population standard deviation is unknown). In general, \(t\)-scores always have the same form:
\[ \text{statistic} = \frac{\text{sample statistic} - \text{assumed population parameter}} {\text{measure of variation of the sample statistic}}. \] The \(t\)-score here is the test statistic, since it is based on sample data (‘a statistic’) and used in a hypothesis test.
A \(t\)-score is similar to a \(z\)-score; both the \(z\)- and \(t\)-scores have the same form: \[ \frac{\text{sample value} - \text{population value}} {\text{measure of variation of the sample value}}. \]
Then:
- If the ‘sample value’ refers to an individual observation \(x\), the measure of variation is the standard deviation, because the standard deviation measures the variation in the individual observations.
- If the ‘sample value’ is a sample statistic, the measure of variation is a standard error, because the standard deviation measures the variation in the sample statistic.