Constructing a Confidence Interval for a Population Mean or Population Mean Difference

A one-sample t interval is $\bar{x} \pm t^{*}\frac{s}{\sqrt{n}}$ with $df = n - 1$. For 25 batteries with $\bar{x} = 124.6$ and $s = 9.2$: $SE = 1.84$, $t^{*} = 2.064$ at 95%, and the interval is about $(120.8, 128.4)$ hours. The conditions are random selection, the 10% condition, and a shape check that is either $n \ge 30$, a stated normal population, or a plot of the sample showing no strong skew or outliers. Paired data collapses first: compute one difference per pair, then run the same interval on those differences with $df$ equal to the number of pairs minus one.

$z^{*} = 1.96$ gets used where $t^{*} = 2.064$ belongs, producing an interval too narrow for its stated confidence, and $df$ gets read at $n$ instead of $n - 1$. The design gets misread, so 12 before-and-after measurements on the same students run as two samples with 22 degrees of freedom instead of one sample of 12 differences with 11. And the referent slips: an interval for $\mu$ reported as a range holding 95% of individual batteries, whose spread of 9.2 hours would make such a range several times wider, or as a probability statement about a parameter that is simply in or out.