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Module 4 · Statistical Inference · INF-3
Last class ended with a debt on the books: Maya used her survey's \(s = 1.3\) as if it were the true \(\sigma\) — forgivable at \(n = 300\). Today the editor wants a follow-up — "do the varsity athletes sleep even worse?" — and between 6 a.m. practices Maya can reach exactly nine of them. Same swap, nine data points. Today: what the shortcut silently breaks, and the 1908 invention that pays the debt properly.
Live session · join with code OTTO-INF3 · ottolearn.org/join
Same rules as always: vote before I explain, every poll is anonymous, and changing your mind after discussion is the goal.
Today · nine athletes, five questions
Why \(z\) with \(s\) breaks at small \(n\) — the label stops being the guarantee
Student's \(t\): heavier tails, one curve per \(df = n - 1\)
The t-table: the right row (\(df\)), the right column (level)
The recipe \(\bar{x} \pm t^* \cdot s/\sqrt{n}\) — license first
The fork: \(\sigma\) known → \(z\); \(\sigma\) estimated → \(t\). Wider = honest.
Watch last class's recipe silently mislabel itself on a small sample, meet the distribution invented in a brewery to fix it, learn to read its table, build the honest interval — and see why the wider answer is the better one. Every step starts with your vote.
Recall · last class · 45 seconds
Warm-up · commit first · 60 seconds · anonymous
Hold your answer — the first poll of the day settles it with a hundred rehearsals.
Commit first · today's summit
Core concept 1 · C1
The recipe ran without complaint — that's the trap. Nothing in \(\bar{x} \pm z^* \cdot s/\sqrt{n}\) warns you that one of its ingredients is now a rough guess.
Same move as last class: the sample's \(s\) stands in for the unknown \(\sigma\). At \(n = 300\), \(s\) hugs \(\sigma\) tightly and the swap is nearly free. At \(n = 9\), \(s\) itself wobbles hard from sample to sample — some nines luck into a small \(s\), others a large one.
\(s \rightarrow \sigma\)The z-bell was computed for a known \(\sigma\). Pretending \(s\) is exact makes the intervals too narrow — they claim more precision than nine points bought, so across many samples they miss \(\mu\) more often than the label admits.
Not a bigger label — a wider yardstick: a distribution with heavier tails that charges for the extra doubt automatically, and charges less as \(n\) grows. It was invented in a brewery, and it's the next slide.
The warm-up, resolved: the swap is not safe at nine — the label stops matching the delivery. First, watch the mislabeling happen a hundred surveys at a time:
See it · C1 · the anchor, resolved
Predict first: at \(n = 6\), 95%, method z — where will the coverage rate settle after many runs? Then I switch the method to t: what should the rate do, and what visibly changes about the intervals?
Core concept 2 · C2
In 1908, Guinness brewer William Gosset faced Maya's exact problem — quality checks on four barrels, not four hundred. The brewery guarded its methods as trade secrets, so he published the fix under a pseudonym: "Student." His distribution is the z-bell's honest cousin.
Same center (0), same symmetry, same bell silhouette — but more probability parked in the extremes. The heavy tails are the fee for estimating \(\sigma\): extreme outcomes genuinely are more likely when your yardstick is itself a guess.
\(t\ \text{vs}\ N(0,1)\)Not one curve — one per degrees of freedom. Few data points → heavy tails (lots of doubt about \(s\)); more data → the tails lighten. Which family member you use is priced by \(df\), and the price list is the next slide.
\(df = n - 1\)As \(n\) grows, \(s\) locks onto \(\sigma\) and the doubt fee shrinks: the t-curve melts into the normal. That's why last class's swap was forgiven — at \(n = 300\), \(t\) and \(z\) are near-twins. At \(n = 9\), they are not.
\(df \to \infty:\ t \to z\)One distribution, built to charge exactly the right fee for a guessed \(\sigma\) — steep at \(n = 9\), nearly free at \(n = 300\). Watch the tails carry the doubt:
See it · C2
Predict first: at \(df = 5\), is the 95% critical value closer to 2.0, 2.6, or 3.2? Then slide \(df\) up — at what point can you no longer tell the two curves apart?
Core concept 3 · C3
The family is indexed by \(df = n - 1\) — and the subtraction isn't a convention, it's an accounting fact: computing \(s\) spends one piece of information before the spread is measured.
To get \(s\), you first compute \(\bar{x}\) — and once \(\bar{x}\) and eight of Maya's nine values are on the table, the ninth is fully determined: the deviations from \(\bar{x}\) must cancel out. Only eight deviations are free to vary.
\(\textstyle\sum (x_i - \bar{x}) = 0\)Nine athletes → \(df = 8\). Sixteen swimmers → \(df = 15\). The degrees of freedom are the row of the t-table you're about to read — always one less than the count.
\(df = n - 1\)The single most common error in this topic: reading row \(n\) instead of row \(n - 1\). Wrong row → wrong \(t^*\) → wrong interval, every time. Subtract before you look. Today's practice items set this trap once each — on purpose.
\(df \neq n\)One subtraction, then a lookup. The lookup is next — the new table has a second dimension, and the row is exactly this \(df\).
Core concepts 4–5 · C4
The normal table needed one number; the t-table needs two. The row is \(df = n - 1\) — how much doubt. The column is the confidence level — how much warranty. The cell where they meet is \(t^*\).
Maya's athletes: \(n = 9\) \(\Rightarrow df = 8\) — row 8, not row 9. The row prices how shaky the estimate \(s\) is: the smaller the \(df\), the bigger every value in the row.
\(df = 9 - 1 = 8\)Same three warranties as last class — 90%, 95%, 99% — one column each. Row 8, column 95%: \(t^* = 2.306\). Compare \(z^* = 1.96\): the nine-point sample pays about 18% more width for the same label.
\(t^*_{8,\ 95\%} = 2.306\)Slide down any column and \(t^*\) shrinks toward the \(\infty\) row — which reads 1.645, 1.96, 2.576: exactly the z-values. The z-recipe is the t-table's basement: the doubt fee fully faded.
\(1.645,\ 1.96,\ 2.576\)Row, column, cell — that's the whole skill. See the lookup run end to end, then drill it until the \(n - 1\) reflex is automatic:
See it · C4
Predict first: I'll set \(n = 9\) at 95%. Which row lights up — 8 or 9? Then, holding the level fixed, what happens to \(t^*\) as I grow \(n\)?
Practice · read the table, solo · fresh numbers on demand
Core concept 6 · C6
Put last class's recipe and today's side by side and only two symbols move — and they move together, for the same reason: \(\sigma\) is now honestly unknown.
The whole architecture: point estimate in the middle, reach out a margin \(E\), print the range, say the claim in words. Everything you learned about intervals last class transfers untouched.
\(\bar{x} \pm E\)\(s\) replaces \(\sigma\) (the data must supply the spread), and \(t^*\) replaces \(z^*\) (the yardstick must charge for that). The two swaps are one package — never mix \(t^*\) with a known \(\sigma\), or \(z^*\) with a nine-point \(s\).
\(z^* \to t^*,\ \ \sigma \to s\)with \(t^*\) read from row \(df = n - 1\) at your chosen level. Same three ingredients as always — center, warranty dial, yardstick — with the dial now reading the t-table instead of the normal table.
\(\bar{x} \pm t^* \cdot \dfrac{s}{\sqrt{n}}\)Watch one assemble first — then I'll build the athletes' interval for real, the one the editor is waiting on.
See it · C6
Predict first: with \(n = 9\) and \(s = 30\), will the t-interval be wider or narrower than the z-version — and by a little or a lot? Then find which control closes the gap.
I do · watch a full build
Watch me decide every step out loud — check the license (shape now matters), price the doubt (\(df \to t^*\)), compute the margin, assemble the claim and say it in words.
Random-ish sample? The nine came from a roster draw, not volunteers — ✓. \(\sigma\)? Honestly unknown: only \(s = 30\) min, computed from the nine — ✓, so \(t\) is required. Shape? Small \(n\) means no CLT rescue, so I check the dot plot: one mound, roughly symmetric, no wild outliers — normality is plausible. License granted.
Nine athletes \(\Rightarrow df = 9 - 1 = 8\) — row 8, not row 9. The paper's standard warranty: the 95% column. The cell: \(t^* = 2.306\). Sanity check before moving on: heavier than \(z^* = 1.96\), exactly as nine-point doubt demands.
Yardstick: \(\text{SE} = s/\sqrt{n}\) \(= 30/\sqrt{9}\) \(= 10\) min. Reach: \(E = t^* \times \text{SE}\) \(= 2.306 \times 10\) \(= 23.06\) min. The z-shortcut would have reached only \(1.96 \times 10 = 19.6\) — I'm buying about 3½ extra minutes of honesty per side.
\(386 \pm 23.06\) \(\Rightarrow (362.94,\ 409.06)\) min — roughly 6.0 to 6.8 hours. In words: "We are 95% confident the athletes' true average sits between 363 and 409 minutes." Now look hard at the top end: 409.
The campus average is 408 minutes. The z-shortcut's interval, \((366.4,\ 405.6)\), excludes it — headline: "athletes sleep less than average." The honest interval includes 408: the data can't back that headline. The shortcut wasn't just narrow — it printed a claim nine data points never earned.
We do · you finish it
New team, new numbers: you supply the margin and the interval — the row changes, the habit doesn't.
Same follow-up, second team. Sixteen swimmers' logs: \(\bar{x} = 402\) min, \(s = 40\) min, \(\sigma\) unknown, dot plot one clean mound. Build the 95% interval.
Finish it: \(E = ?\) — and the interval is…? (Then check it against the campus 408.)
Practice · build the t-interval, solo · fresh numbers on demand
Now you build whole t-intervals solo — fresh numbers every re-roll, the table excerpt in every item, support at its minimum.
Core concept 7 · C7
The t-distribution pays for a guessed \(\sigma\) — it does not pay for bad sampling or a wild population shape. Three boxes to tick, every time:
Independence comes from the draw, not from the data. Nine volunteers who "felt like sharing their sleep" break the license before any formula runs — the same rule since the very first survey.
The tell: \(s\) was computed from the sample. (If \(\sigma\) genuinely is known — a machine with years of records — the z-recipe stays exactly correct at any \(n\). That fork is two slides away.)
Small \(n\) means no CLT rescue — the population itself must look roughly normal. Check the picture: one mound, rough symmetry, no extreme outliers.
License granted → build. License denied (strong skew, extreme outliers at tiny \(n\)) → collect more data, or report honestly that no interval is defensible yet. Now, the everyday fork:
Core concept 8 · C8
C7 was the stop sign: may we build at all? This is the fork just past it: which yardstick? One question decides — and it is not "is \(n\) big?"
Years of machine records, a published spec — genuine outside knowledge of \(\sigma\). Then the z-recipe is exactly right at any \(n\), even \(n = 9\). (Rare outside factories and labs.)
\(\bar{x} \pm z^* \cdot \sigma/\sqrt{n}\)If \(s\) came from the data, use \(t\) — always, at every \(n\). This is most of real life: nobody hands a reporter the true \(\sigma\). The only question left is which row: \(df = n - 1\).
\(\bar{x} \pm t^* \cdot s/\sqrt{n}\)At \(n = 300\), \(t^* \approx 1.968\) vs \(z^* = 1.96\) — last class's shortcut was a rounding error, which is why it was forgiven. \(t\) stays the principled choice; \(z\) is its acceptable big-\(n\) approximation.
\(df \uparrow:\ t^* \to z^*\)The decision runs on information, not on sample size: known \(\sigma\) → \(z\); estimated \(\sigma\) → \(t\). See the whole map, then vote it:
See it · C8
Predict first: where does "\(n = 45\), \(\sigma\) unknown" land — and does the answer change at \(n = 9\)? Find the one region where \(z\) is legitimate at small \(n\).
Which tool applies? · mixes earlier classes + today
Practice · pick the yardstick, solo · fresh numbers on demand
Core concept 9 · C9
New table, same contract. Everything the opening poll rehearsed about "95% confident" applies to t-intervals word for word.
"We are 95% confident the athletes' true average sleep is between 363 and 409 minutes." — the same accepted shorthand as last class, crediting the method that built the interval.
A statement about the procedure: across many nine-athlete studies, about 95% of t-intervals built this way capture \(\mu\). The t-table's entire job was making that promise true again at small \(n\).
Not a probability on \(\mu\) (a fixed constant), not a range for individual athletes, not a promise about future samples. Every misread from last class stays a misread — no new ones today.
That's the quiet payoff of the whole lesson: \(t\) exists so that the printed warranty means what it says — at \(n = 9\), not just \(n = 300\).
Core concept 10 · C10
Two intervals now sit on Maya's desk, built from the same nine logs. Instinct says print the tighter one. Instinct is wrong.
At \(df = 8\): \(t^* = 2.306\) vs \(z^* = 1.96\) — about 18% more width. The gap is the doubt about \(s\), priced. It shrinks as \(n\) grows and never quite reaches zero.
\(t^* > z^*\)The z-interval \((366.4,\ 405.6)\) looks better — tighter, more decisive, and it comes with a headline. But its 95% label is unearned: it's the interval that misses \(\mu\) more often than it admits.
The t-interval \((362.94,\ 409.06)\) is wider because it isn't hiding anything. When \(\sigma\) is guessed, honest uncertainty is wider uncertainty. Precision you didn't pay for is confidence theater.
A narrower interval is only better if its label is true. One last vote to make it stick:
Quick check · which one gets printed?
Production task · the follow-up she files
Exit ticket · 60 seconds · anonymous
What Maya files
The IOU from last class is settled: when \(s\) stands in for \(\sigma\), the yardstick must charge for the doubt — Student's \(t\), row \(df = n - 1\). At \(n = 300\) the fee was invisible; at \(n = 9\) it was the whole story.
The honest interval — \((362.94,\ 409.06)\) minutes — kept its promise by being wider, and it quietly killed a false headline the z-shortcut would have printed. What Maya files is a truthful non-finding: nine athletes can't yet distinguish the team from the campus.
And the fork is permanent equipment now: \(\sigma\) genuinely known → \(z\) at any \(n\); \(\sigma\) estimated from data → \(t\) at any \(n\), melting into \(z\) as samples grow.
Next on the desk: a reader asks not about hours but about a fraction — "what percentage of students pull all-nighters?" A yes/no answer per student, a sample proportion… and a familiar recipe with one more new yardstick. That's next class.
Vote counts on today's slides are simulated for rehearsal; live sessions show the room's real votes.