Choosing z or t, and the CI checkpoint

The decision

Every interval for a mean in this course is built the same way once one question is settled: which critical value? The decision has two branches:

1. Is $\sigma$ given? If the question supplies the population SD — from history, from the manufacturer, "known to be" — the critical value is $z_{\alpha /2}$, whatever $n$ is.
2. Otherwise, $s$ is estimated from the sample. The distribution is $t$ with $n-1$ df. If $n-1\le 30$, read that row; if $n-1>30$, read the $\infty$ row, which is $z$ again — but the reason is the large-df rule, not a known $\sigma$.

The branch is taken on whether $\sigma$ is known, first, and on $n$ second. The classic mistake is to run the branches in the wrong order: "$n=60$ is large, so $z$" is right by accident when $\sigma$ is unknown and wrong when it is the reason given. Say the reason.

Worked example — three cases, one grammar

Cedar & Stone reports a sample mean of $\bar{x}=88.4$ minutes for the time to prepare a tax return, with sample SD $s=6.3$, at 95 %.

Case A — $\sigma =6.3$ is a known figure from years of records, $n=12$. Known $\sigma$ → $z=1.96$. $\mathrm{SE}=6.3/\sqrt{12}=1.819$, margin $3.56$, interval $(84.84,91.96)$.

Case B — $s=6.3$ from the sample, $n=12$. Unknown $\sigma$, df $=11\le 30$ → $t_{0.025;11}=2.201$. Same SE $1.819$, margin $2.201\times 1.819=4.003$, interval $(84.40,92.40)$. Wider than A by 12 %: the cost of estimating $\sigma$ from eleven degrees of freedom.

Case C — $s=6.3$ from the sample, $n=60$. Unknown $\sigma$, df $=59>30$ → $\infty$ row, 1.96. $\mathrm{SE}=6.3/\sqrt{60}=0.813$, margin $1.96\times 0.813=1.594$, interval $(86.81,89.99)$. The multiplier is $z$'s value, but the sentence on the paper is t with 59 df; ∞ row by the large-df rule.

Cases B and C share $\bar{x}$ and $s$; the fivefold larger sample cuts the width from 8.0 to 3.2 minutes — the SE fell by $\sqrt{5}$ and the multiplier fell from 2.201 to 1.96.

Then the same three lines

Whichever branch, the working is:

• $\mathrm{SE}=\sigma /\sqrt{n}$ or $s/\sqrt{n}$;
• margin $=$ critical $\times$ SE;
• interval $=\bar{x}\pm$ margin, reported as two limits with units.

And the sentence: we are 95 % confident that the mean preparation time is between 84.40 and 92.40 minutes. Conditions on the side: an SRS, and a normal population or $n>30$ — with a note on skew and outliers when $n<15$.

Where this sits

This decision is one branch of the selection tree that Module 14 assembles for the whole course: one mean → σ known or not → $z$ or $t$. Module 10 reuses it for tests (the same choice picks a z-test or a t-test), Module 11 extends it to two means, Module 12 to proportions (always $z$), Module 13 to a slope (always $t$). Learn this branch cold and the rest are variations.