Two-Sample t Intervals and Reading t Output

The interval for a difference in means

With two independent random samples, the confidence interval for $\mu_1-\mu_2$ is

Conditions: both samples random; independent of each other; each population normal or each sample size at least $30$ (check a dotplot of each sample for strong skew or outliers when $n$ is small); and the $10\%$ condition for each. AP Statistics does not pool the two variances — the formula above is the only one used.

Which degrees of freedom?

Technology computes a fractional $df$ (the Welch formula), typically between $\min(n_1-1,n_2-1)$ and $n_1+n_2-2$. By hand, the conservative choice is $df=\min(n_1-1,\,n_2-1)$: it gives a slightly larger $t^*$ and a slightly wider interval, which is acceptable on the exam as long as you state it. Never use $df=n_1+n_2-2$; that belongs to the pooled procedure.

Reading output and interpreting

Calculator or software output lists the difference, the standard error, $df$, and either an interval or a $t$ with its P-value. Example: $n_1=25$, $\bar x_1=72$, $s_1=8$; $n_2=30$, $\bar x_2=66$, $s_2=10$. The standard error is $\sqrt{64/25+100/30}=\sqrt{5{,}89}\approx2{,}43$; with $df=24$, $t^*=2{,}064$ for $95\%$, giving $6\pm5{,}0$, or $(1{,}0;\,11{,}0)$. Interpretation: "we are $95\%$ confident that the true mean of group 1 exceeds that of group 2 by between $1{,}0$ and $11{,}0$ units". An interval that excludes $0$ is evidence of a difference; one that includes $0$ is not.