In the discussion that follows, we shall assume that we are sampling from a normal population. We determined above that a confidence interval for the population mean μ is when σ is known. But if σ is unknown, it must be estimated by
Additionally, if n is “small” , then the statistic
follows a t distribution withdegrees of freedom. So once s replaces σ and n is small, the resulting statistic [Eq. (8.10)] is no longer N(0, 1).
Let us examine the properties of the sampling distribution of the t statistic [Eq. (8.10)]: