- Check the conditions for a [latex]t[/latex]-distribution, then use a [latex]t[/latex]-distribution to calculate probabilities when appropriate.
[latex]t=\dfrac{\stackrel{¯}{x}-μ}{SE(\stackrel{¯}{x})} = \dfrac{\stackrel{¯}{x}-μ}{\frac{s}{\sqrt{n}}}[/latex]
will follow a [latex]t[/latex]-distribution with [latex]n-1[/latex] degrees of freedom if the population distribution is normal or if the population distribution is not too skewed and the sample size is large (e.g., [latex]n \ge 30[/latex]).
Since the [latex]t[/latex]-statistic exhibits more sampling variability than the [latex]z[/latex]-statistic, its distribution has slightly more variability than a standard normal distribution. However, as the sample size increases, there is less sampling variability associated with the standard error of the sample mean, so its distribution gets closer to a standard normal distribution.