Section 9.4 Summary

1. The test statistic used for the $t$ test for the mean is $t_{\text{data}}=\frac{\overline{x}-{\mu }_0}{s/\sqrt{n}}$

   with $n-1$ degrees of freedom. The $t$ test may be used under either of the following conditions: (a) the population is normal, or (b) the sample size is large $\left(n\ge 30\right)$. For the critical-value method, we compare the values of $t_{\text{data}}$ and $t_{\text{crit}}$. If $t_{\text{data}}$ falls in the critical region, we reject $H_0$.

2. For the $p$-value method, we reject $H_0$ if the $p\text{-value}\le \alpha$.
3. We may use $100\left(1-\alpha \right)\%$$t$ confidence interval to perform two-tailed $t$ tests at level of significance $\alpha$ for various values of ${\mu }_0$.