Inference for Quantitative Data: Means

Notes

Why $t$ instead of $z$

We rarely know $\sigma$, so we estimate it with $s$. The standardized statistic then follows a $t$ distribution with $df = n - 1$. $t$ distributions are symmetric and bell-shaped but have heavier tails than the normal; as $df$ grows they approach the normal.

Conditions for means

• Random: random sample or random assignment.
• 10%: $n \le 10\%$ of the population when sampling without replacement.
• Normal/large sample: $n \ge 30$, or the population is approximately normal, or a graph of the sample shows no strong skew or outliers.

One-sample $t$ procedures

• Interval: $\displaystyle \bar x \pm t^*\frac{s}{\sqrt n}$.
• Test: $\displaystyle t = \frac{\bar x - \mu_0}{s/\sqrt n}$, with $df = n - 1$.
• Paired data (two measurements on the same individual, like before and after): take the differences, then do one-sample $t$ on $\mu_{diff}$.

Two-sample $t$ procedures

For two independent groups:

• Interval: $\displaystyle (\bar x_1 - \bar x_2) \pm t^*\sqrt{\frac{s_1^2}{n_1} + \frac{s_2^2}{n_2}}$.
• Test of $H_0: \mu_1 = \mu_2$: $\displaystyle t = \frac{(\bar x_1 - \bar x_2) - 0}{\sqrt{\frac{s_1^2}{n_1} + \frac{s_2^2}{n_2}}}$. Use technology for $df$ (or the conservative smaller $n - 1$).