Week 11 · Lecture outline

Week 11 — Lecture Outline · Confidence Intervals for a Mean

Introduction to Statistics Generic evergreen edition

Course: Introduction to Statistics (18-week generic edition)
Objectives covered: Objective 6 — Construct and interpret confidence intervals for a population mean (the mean portion; proportions follow in Week 12).
SLOs touched: A (reason quantitatively from data) · B (communicate results to a non-technical audience)
Meeting pattern: planned as 2 sessions × ~75 min = ~150 min. Segment minutes below total ~150; scale them to your own pattern.

Week at a Glance


The week's big question	"A sample gives one number — but the truth is a number we'll never see. How do we turn one honest sample into a range we can defend, with a confidence we can state?"
By the end of the week, students can…	(1) name the point estimate for a mean (x̄) and say why a bare point estimate is almost surely wrong; (2) explain why t replaces z when σ is unknown, and find degrees of freedom (df = n − 1) and t* from the course's friendly t-table; (3) construct a one-sample t-interval — check conditions, compute SE = s/√n, ME = t·SE, x̄ ± ME; (4) take a margin of error apart* and predict how confidence level, n, and s change the width; (5) interpret a confidence interval correctly — and catch the two classic misreads.
Key vocabulary	point estimate, confidence interval, confidence level, margin of error (ME), standard error (SE = s/√n), t-distribution, degrees of freedom (df), critical value (t*), one-sample t-interval, conditions (random, roughly normal for small n), capture rate, plausible values
Materials	slides (Deck 11), the Week 11 chapter (with the friendly t-table), the week's readings + video links, a spreadsheet (Google Sheets or Excel), the student's chatbot for the AI-critique moment and the tutorial
Timing note	8 segments, ~150 min total. Session 1 = Segments 1–4 (~75). Session 2 = Segments 5–8 (~75).

Segment 1 — Hook & the Promise (8 min) · Session 1 opens

Hook. "Your sleep app announces: you average 6.9 hours a night. One number, two decimal places, total confidence. Here's the question nobody asks the app: how wrong could that number be? It watched a few weeks, not your whole life — so 6.9 is a sample's answer, and Week 10 taught us exactly what sample answers do: they wobble."

Every "average" you've ever been quoted — battery life, delivery windows, calories, wait times — was a point estimate: one sample's best guess, dressed up as the truth.
"This week we stop bluffing. Instead of one number that's almost surely wrong, we'll report a range we can defend — and attach an honesty label to it: how confident we are, and in what sense."

The promise (write it on the board): "By the end of this week you can take any small sample — ten laptops, sixteen coffee pours, twenty-five checkout lines — and produce a range for the true mean, state exactly how confident you are, and catch the two ways almost everyone misreads that sentence."

Why it matters line (memory hook): "An honest estimate is a range with a confidence attached — one number is a bluff."

Segment 2 — Point Estimates & the Interval Idea (20 min)

Plain language first.
- A point estimate is the single best guess a sample can give: for a population mean μ, it's the sample mean x̄. (P→P, S→S from Week 1: μ is the parameter we want; x̄ is the statistic we have.)
- The point estimate's problem is not that it's a bad guess — it's that, as a single point, it is almost surely not exactly μ. Week 10's sampling-distribution engine says so: x̄ lands near μ, wobbling with standard error SE = s/√n, and essentially never on it.
- The fix is ancient and honest: report the guess ± some wiggle room. That's a confidence interval (CI):

CI = point estimate ± margin of error — "the guess, plus or minus how much guesses like this typically miss by."

The margin of error (ME) will be built from two parts: how much x̄ wobbles (the SE, from Week 10) × how many wobbles of cushion we want (a multiplier set by the confidence level — 90%, 95%, 99%).

Callback (make it explicit): Week 10 showed that x̄'s typical miss is σ/√n — and that σ vs. σ/√n (SD vs. SE) distinction is the exact engine here. New this week: we don't know σ, so we use the sample's s — and that substitution is what Segment 3 is about.

Misconception + cure:
- ❌ "The sample mean is wrong, so the sample failed."
✅ Cure: an estimate isn't a prophecy. Missing by a predictable amount is the success — the whole method runs on knowing the typical miss. The failure would be pretending there's no miss at all.

Segment 3 — Why Not z? The t-Distribution & the Friendly Table (25 min)

Plain language first. Week 8's normal machinery used z — which assumes we know the population SD σ. In real life nobody hands you σ. We estimate it with the sample's s — and that estimate is itself wobbly, especially in small samples. Using z anyway would mean ignoring that extra wobble and acting more certain than we are.

The honest fix is a slightly different curve: the t-distribution — bell-shaped, centered at 0, but with heavier tails than the normal. The heavy tails are the price of estimating σ: rare-but-large misses are a bit more common, so the curve budgets for them.
There's a different t curve for every sample size, indexed by degrees of freedom: df = n − 1. Small n → heavier tails → bigger multipliers. As n grows, s pins down σ better, and t melts into z.

Memory hook: "t is z with humility." (And: df = n − 1 — "one degree spent estimating the mean.")

THE FRIENDLY t-TABLE (embedded in the chapter, tutorial, and assignment coach — every problem this week lands exactly on it):

The table below gives the two-sided critical value t* for the three sample sizes this week uses, plus the z* row t approaches.

df (n)	90%	95%	99%
9 (n = 10)	1.833	2.262	3.250
15 (n = 16)	1.753	2.131	2.947
24 (n = 25)	1.711	2.064	2.797
z* (very large n)	1.645	1.960	2.576

Walk the table out loud (three reads):
- Down a column: at 95%, t* falls 2.262 → 2.131 → 2.064 → 1.960. More data → less humility needed → t* slides toward z*.
- Across a row: at df 24, 1.711 → 2.064 → 2.797. More confidence costs a bigger multiplier.
- Corner check: small sample + high confidence (df 9, 99%) = the biggest number on the table, 3.250. Makes sense: little data and big promises require the widest cushion.

Worked example (do every step out loud):

A reviewer will test n = 10 laptops → df = 10 − 1 = 9 → for a 95% interval, t* = 2.262. A café will sample n = 16 pours → df = 15 → 95% gives t* = 2.131. Note both beat z* = 1.960 — t is always a bit wider than z at the same confidence; that's the humility tax.

Misconceptions + cures:
- ❌ "df = n."
✅ Cure: df = n − 1 — one degree of freedom is spent estimating x̄ before s can be computed. n = 16 means row df 15, not 16.
- ❌ "Just use 1.96; it's close enough."
✅ Cure: at n = 10, the honest multiplier is 2.262 — using 1.96 shaves ~13% off your margin and quietly overstates your certainty. (This is also the #1 chatbot error this week — Segment 8 makes students catch it.)

Segment 4 — Misconceptions Checkpoint + Think-Pair-Share (22 min) · Session 1 closes (~75)

Name the misconceptions out loud, then cure each:

❌ "SE and SD are the same thing."
✅ Cure (Week 10's drum, beaten again): s describes individuals' spread; SE = s/√n describes how much the sample mean wobbles. The interval is built on SE — using s in its place produces an absurdly wide interval (√n times too wide).
❌ "A confidence interval is a range of the data."
✅ Cure: the interval hunts μ, the mean — it says nothing about where individual values fall. Most individuals will be outside a good interval for the mean; that's normal (individuals wobble by s, means by s/√n).
❌ "Bigger t* would be better — more confidence!"
✅ Cure: confidence is bought with width. A 99.999% interval is nearly certain and nearly useless ("your mean sleep is between 2 and 12 hours"). Precision and confidence trade off; you choose the balance.
❌ "n = 16, so look up df 16."
✅ Cure: df = n − 1. Say the subtraction out loud every time until it's automatic.

Interaction — Think-Pair-Share (rapid-fire, ~10 min). Six quick items on a slide; solo 30 sec, pair 1 min, fingers vote. (Table on screen.)
1. A sample of n = 10 → df = ? → 9
2. t* for a 95% interval with n = 25 → 2.064
3. s = 10, n = 25 → SE = ? → 10/√25 = 2
4. t* = 2.131 and SE = 2 → ME = ? → 2.131 × 2 = 4.262
5. Same data — which is wider, a 90% or a 99% interval? → 99%
6. True or false: about 95% of individual values fall inside a 95% CI for the mean → False (it's an interval for the mean).

Debrief items 3 and 6 — the SE-vs-SD divide and the individuals-vs-mean divide are the two walls students hit all week.

Segment 5 — Building the t-Interval: Conditions + the Recipe (25 min) · Session 2 opens

Hook back in: "Last session bought the parts: SE for the wobble, t* for the humility. Today we assemble the machine."

First, the conditions (the password before the formula):
1. Random: the data are a random sample (or reasonably like one) from the population — Week 1's lesson never retires; no formula rescues a biased sample.
2. Roughly normal (matters for small n): for n around 25 or less, the data shouldn't be strongly skewed or carry wild outliers — check a quick histogram/dot plot. For larger n, Week 10's CLT does the heavy lifting.

The recipe (drill it as four beats): ① check conditions → ② SE = s/√n → ③ ME = t* × SE (df = n − 1, pick the confidence column) → ④ interval = x̄ ± ME. Then say it in words.

Worked example 1 — the week's centerpiece (do every step out loud):

A wellness program records nightly sleep for a random sample of n = 25 members: x̄ = 7.2 hours, s = 1.0 hour. Build a 95% confidence interval for the mean nightly sleep of all members.
- ① Random sample given; n = 25 with no strong skew reported — conditions pass.
- ② SE = 1.0/√25 = 1.0/5 = 0.2 hours — the typical wobble of a 25-person mean.
- ③ df = 24; 95% → t* = 2.064; ME = 2.064 × 0.2 = 0.4128 ≈ 0.41 hours.
- ④ Interval: 7.2 − 0.41 to 7.2 + 0.41 → (6.79, 7.61) hours (unrounded: 6.7872 to 7.6128 — round at the end, not in the middle).
- In words: "We are 95% confident the mean nightly sleep of all members is between about 6.8 and 7.6 hours."

Worked example 2 — smaller sample, different df (every step):

A café's espresso machine pours a "355 mL" house coffee. A barista randomly pulls n = 16 cups: x̄ = 354 mL, s = 8 mL. 95% interval for the true mean fill:
- SE = 8/√16 = 8/4 = 2 mL · df = 15 → t* = 2.131 · ME = 2.131 × 2 = 4.262 mL.
- Interval: 354 ± 4.262 → (349.7, 358.3) mL.
- Read it like an analyst: the label's 355 sits inside the interval — the machine's true mean fill being 355 is plausible; this sample gives no grounds to accuse the machine. (Whisper the Week 13 tease: "this inside/outside move is hypothesis testing wearing pajamas.")

Misconception + cure:
- ❌ Rounding SE or t*·SE early, then wondering why endpoints don't check out.
✅ Cure: carry full digits to the final step; round only the endpoints (and say how much you rounded).

Segment 6 — Margin-of-Error Anatomy: What Moves the Width (18 min)

Plain language first. ME = t* × s/√n. Three dials, three predictable effects:

Confidence level ↑ → wider. More certainty costs more width (bigger t*).
n ↑ → narrower. More data shrinks SE — but by √n, so quadrupling n only halves the margin. (Precision is bought at a steep exchange rate.)
s ↑ → wider. Noisier populations give wobblier means. You don't choose s; you inherit it.

Worked example — same data, three confidence levels (the slide students photograph):

Sleep data again (x̄ = 7.2, SE = 0.2, df = 24):
- 90%: ME = 1.711 × 0.2 = 0.3422 → (6.86, 7.54) — narrowest, least sure.
- 95%: ME = 2.064 × 0.2 = 0.4128 → (6.79, 7.61) — the convention.
- 99%: ME = 2.797 × 0.2 = 0.5594 → (6.64, 7.76) — surest, widest.
One dataset, three honest reports — differing only in how much certainty was purchased. Certainty costs width.

Reverse-reading drill (teach it now; the assignment uses it): given a finished interval, recover the pieces — center = (low + high)/2 = x̄, ME = (high − low)/2. An interval of (83.7, 100.3) has x̄ = 92 and ME ≈ 8.3. Any reported interval can be taken apart with two subtractions.

Misconception + cure:
- ❌ "To halve the margin, double the sample."
✅ Cure: √n is in the basement — halving ME takes 4× the sample. This is why precise studies are expensive, and why tiny-sample precision claims should raise your eyebrow.

Segment 7 — Interpretation: What "95% Confident" Buys You (20 min)

Plain language first. The interval (6.79, 7.61) is built. Now the harder skill: saying what it means without lying.

The correct read: "We are 95% confident that the mean nightly sleep of all members is between 6.79 and 7.61 hours."
What "95%" actually describes: the METHOD. If many teams each drew their own random sample and built their own interval this way, about 95% of those intervals would capture the true μ — and about 1 in 20 would honestly miss. Your interval is one ticket from that batch: probably a winner, possibly not, and you can't tell which from the inside.

Worked example (checkout times — set up the misread hunt):

A supermarket times a random n = 25 self-checkout transactions: x̄ = 92 s, s = 20 s → SE = 20/√25 = 4 s → ME = 2.064 × 4 = 8.256 → 95% CI (83.7, 100.3) seconds.
Now police three student sentences:
- ❌ "95% of customers finish between 83.7 and 100.3 s." — No: that's individuals; the interval is about the mean. (Individual times spread with s = 20; plenty finish in 60 or 130 s.)
- ❌ "There's a 95% probability μ is in (83.7, 100.3)." — Gently no: μ is a fixed number, not a spinner; it's in this interval or it isn't. The 95% lives in the procedure — 19 in 20 such intervals capture μ. (Police the phrasing, not the intuition: "95% confident" is the licensed shortcut.)
- ✅ "We're 95% confident the store's true mean checkout time is between about 84 and 100 seconds — a range produced by a method that captures the truth 19 times in 20."

Second read — intervals as plausible values: every value inside the interval is a claim the data can live with; values outside are implausible. "Is the true mean checkout time 100 s?" — 100 is (barely) inside: plausible. "Is it 110?" — outside: the data push back. This framing is the on-ramp to Week 13.

Misconception + cure:
- ❌ "Our interval captured μ — I checked, the middle looks right."
✅ Cure: nobody ever knows whether their interval captured μ — that's the point of the confidence statement. (This week's data lab makes the class see it: everyone builds an interval from their own penguin sample around a knowable full-data mean, and roughly 95% of the room captures it — a few honest intervals miss.)

Segment 8 — Technology Workflow + AI-Critique, Callback & Hand-off (12 min) · Session 2 closes (~75)

Technology workflow — the whole interval in a spreadsheet (exact steps, live):
1. Sample values in A2:A26 (n = 25). Then: =AVERAGE(A2:A26) → x̄ · =STDEV(A2:A26) → s · =STDEV(A2:A26)/SQRT(25) → SE.
2. The multiplier, from software instead of our table: =T.INV(0.975, 24) → 2.0639 — the same 2.064 as the friendly table (0.975 because 2.5% sits in each tail; df = 24). Google Sheets and Excel are identical here.
3. One-liner for the margin: =CONFIDENCE.T(0.05, 1.0, 25) → 0.4128 for the sleep data (alpha = 0.05, s, n) — matching our hand-built ME. Endpoints: x̄ − ME and x̄ + ME.
4. Sanity habit: does the interval contain x̄ dead-center? Is ME bigger than SE? If not, something's mis-typed.

AI-critique moment (students verify, not consume):

Paste this to your chatbot: "A random sample of 10 laptops has mean battery life 9.5 hours with standard deviation 1.5. Build a 95% confidence interval for the mean. Show the multiplier you used."
Then check its work against the friendly table: the honest multiplier is t* = 2.262 (df 9). Chatbots routinely grab z* = 1.96 — an interval ~13% too narrow that quietly overstates certainty — or quote a t-table "from memory" and drift by a digit. If its multiplier isn't 2.262, you've caught it. The tool drafts, you judge. This exact catch is graded in this week's lab.

Callback + tease:
- Callback: "Week 10 built the engine — sample means wobble by a knowable SE. This week we drove it somewhere: from one sample to a defensible range for the truth."
- Tease next week: "Same machine, new fuel: Week 12 builds confidence intervals for proportions — and finally decodes the '±3 points' fine print under every poll you've ever read."

Hand-off (the week's work):
- Chapter 11 (primary reading, with the friendly t-table) — then Lecture Tutorial 11 (AI tutor; share link + summary).
- Data Lab 11 (your own penguin interval — does the class hit ~95%?) · Quiz 11 (end of week) · Discussion 11 ("The Single-Number Bluff") · Assignment 11 (AI-coached).

Instructor FAQ — Common Stumbles

Student says / does	Quick cure
Uses s where SE belongs (interval √n too wide).	The interval tracks the mean's wobble: SE = s/√n. Quick tell: a "±1.0 hour" margin from 25 people should smell wrong — means are steadier than people.
Looks up df = n instead of n − 1.	df = n − 1, one degree spent estimating x̄. n = 16 → row 15. Have them say the subtraction aloud every time.
"Why not just use 1.96?"	Because σ is estimated, not known. At n = 10 the honest multiplier is 2.262 — 1.96 fakes precision. t melts into z as n grows; the table shows the melt.
"95% of customers are in the interval."	The interval is for the mean, not individuals. Individuals spread by s; most sit outside a good CI for μ.
"There's a 95% chance μ is in my interval."	μ doesn't move — the method has a 95% capture rate: 19 of 20 such intervals catch it, and no one knows which they hold. "95% confident" is the licensed phrasing.
Rounds SE mid-calculation, endpoints don't reconcile.	Carry digits to the end; round only the final endpoints and say so ("≈ 0.41").
"Doubling n halves the margin, right?"	√n: halving ME costs 4× the data. Show 25 → 100 shrinking SE 0.2 → 0.1.
"My data are skewed with n = 12 — can I still do this?"	Small-n t needs roughly normal data: strong skew/outliers → the interval's honesty label is void. Options: bigger sample, or flag the caveat loudly.
"Which confidence level is 'right'?"	None — it's a purchasing decision: certainty costs width. 95% is convention, not law; what matters is saying which you bought.

Scope flag

This outline stays within Objective 6's mean portion. The T.INV/CONFIDENCE.T spreadsheet functions, the "plausible values" framing (a deliberate Week 13 on-ramp), and the capture-rate class experiment framing are added context (not strictly required by the objective) — kept because they cure this week's misconceptions; cut them for a leaner session. Choosing a sample size for a target margin is deferred to Week 12 (proportions), where the algebra is cleaner.