Section 8.2 Summary

1. For a normal population, the distribution of
   $t={\displaystyle \frac{\overline{x}-\mu }{s/\sqrt{n}}}$

   follows a $t$ distribution, with $n-1$ degrees of freedom, where $\overline{x}$ is the sample mean, $\mu$ is the unknown population mean, $s$ is the sample standard deviation, and $n$ is the sample size. The $t$ distribution is symmetric about its mean 0, just like the $Z$ distribution. However, the $t$ distribution is flatter.

2. A $100(1-\alpha )\%$ confidence interval for $\mu$ is given by the interval
   $\overline{x}\pm t_{\alpha /2}(s/\sqrt{n})$

   where $\overline{x}$ is the sample mean, $t_{\alpha /2}$ is associated with the confidence level and $n-1$ degrees of freedom, $s$ is the sample standard deviation, and $n$ is the sample size. We can construct a $t$ interval whenever either of the following conditions is met: the population is normal, or the sample size is large ($n\ge 30$).