Paired data is one sample of differences

$$\bar{x} \pm t^{*}\frac{s}{\sqrt{n}}, \qquad df = n - 1.$$

The critical value $t^{*}$ is read from the t distribution with $n - 1$ degrees of freedom at the stated confidence level, and it is larger than the corresponding $z^{*}$ at every finite df.

A random sample of 25 batteries gives $\bar{x} = 124.6$ hours and $s = 9.2$ hours. Then $SE = \frac{9.2}{\sqrt{25}} = 1.84$, $df = 24$, and $t^{*} = 2.064$ at 95% confidence, so

$$124.6 \pm 2.064(1.84) = 124.6 \pm 3.80,$$

giving roughly $(120.8, 128.4)$ hours.

Two failures live in that line. Using $z^{*} = 1.96$ instead of $t^{*} = 2.064$ produces $124.6 \pm 3.61$, an interval too narrow to deliver its stated 95%. And reading $t^{*}$ at $df = 25$ rather than 24 picks a neighbouring member of the family; the error is small here and grows quickly as $n$ falls.

Three conditions, with the third differing from the proportion case:

1. Random: a random sample or randomized experiment.
2. 10%: $n \le 0.10N$ when sampling without replacement.
3. Normal or large enough: either the population is stated to be approximately normal, or $n \ge 30$, or the sample's own plot shows no strong skew and no outliers.

The third condition is the one that needs judgment rather than arithmetic. At $n \ge 30$ the Central Limit Theorem covers ordinary departures from normality. Below 30, the sample itself is the evidence available, so a dotplot or boxplot is drawn and described: "the dotplot of the 12 measurements is roughly symmetric with no outliers, so the t procedure is reasonable."

Two opposite errors follow from skipping the plot. Refusing inference at $n = 60$ because the population "might not be normal" ignores the CLT, and invoking the CLT at $n = 6$ asks it to do work it cannot. Both are avoided by looking at the data.

When each individual, or each matched pair, produces two measurements, the two columns are not two samples. The correct procedure has three steps:

1. Compute a difference for each pair, in a stated order, so 12 students give 12 differences.
2. Treat those differences as one sample: compute $\bar{x}_d$ and $s_d$.
3. Run the ordinary one-sample t interval on them, with $df = n_{\text{pairs}} - 1$.

The parameter is then $\mu_d$, the mean difference in the population, and the interval estimates it. For 12 students measured before and after a program, $df = 11$, not 22: the sample size is the number of pairs.

Recognizing the design is the whole skill. Before-and-after on the same subjects, twins split across treatments, left hand versus right hand, matched pairs by age and sex: all paired. Two unrelated groups of different individuals: two independent samples, which is Topic 4.7. Running paired data as two independent samples throws away the pairing that made the comparison precise, and running two independent groups as pairs is not even computable unless the group sizes happen to match.

"We are 95% confident that the interval from 120.8 to 128.4 hours captures the true mean lifetime of all batteries of this type."

Three wrong referents, each attaching the interval to the wrong object:

1. "95% of batteries last between 120.8 and 128.4 hours." That describes individuals, whose spread is 9.2 hours; a range holding 95% of individual batteries would be several times wider.
2. "95% of future sample means will fall in this interval." That describes statistics, not the parameter.
3. "There is a 95% probability that $\mu$ is in this interval." The parameter is fixed; it is in or out.

For paired data the sentence names the mean difference and the order of subtraction: "we are 95% confident the interval captures the true mean improvement, after minus before, for students in this program." Dropping the word difference turns a claim about change into a claim about a level.