3.3.3 The $t$ distribution as a solution to the standard error problem

When estimating the mean and standard error from a small sample, the $t$ distribution is a more accurate tool than the normal model. This is true for both small and large samples.

To proceed with the $t$ distribution for inference about a single mean, we must check two conditions.

Independence of observations.

We verify this condition just as we did before. We collect a simple random sample from less than 10% of the population, or if it was an experiment or random process, we carefully check to the best of our abilities that the observations were independent.

Observations come from a nearly normal distribution.

This second condition is difficult to verify with small data sets. We often (i) take a look at a plot of the data for obvious departures from the normal model, and (ii) consider whether any previous experiences alert us that the data may not be nearly normal.

When examining a sample mean and estimated standard error from a sample of $n$ independent and nearly normal observations, we use a $t$ distribution with $n-1$ degrees of freedom ($df$). For example, if the sample size was 19, then we would use the $t$ distribution with $df=19-1=18$ degrees of freedom and proceed exactly as we did in Chapter 2.6, except that now we use the $t$ distribution, i.e. pt or qt in R.