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Week 14 · Lecture outline

Week 14 — Lecture Outline · Testing Claims About Means

Introduction to Statistics Generic evergreen edition

Course: Introduction to Statistics (18-week generic edition)
Objectives covered: Objective 7 — Conduct and interpret hypothesis tests (the means portion: one-sample t-test and paired-data t procedures; proportions and two-sample tests follow in Week 15). Reinforces Objective 6 (test ↔ interval duality).
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 company promises an average — '50 minutes,' '40 mpg.' Your sample disagrees a little. How far from the promise does the data have to drift before 'a little off' becomes 'we don't believe you'?"
By the end of the week, students can… (1) run a complete one-sample t-test — hypotheses, conditions, t-statistic, verdict — and state the conclusion in context; (2) choose between a one-sided and two-sided alternative before seeing the data, and explain how the choice changes the verdict; (3) recognize paired data and analyze it as a one-sample t-test on the differences; (4) use the test ↔ interval duality: a value inside the 95% CI survives the two-sided test at α = 0.05; (5) check the conditions before trusting any of it.
Key vocabulary one-sample t-test, test statistic t, standard error s∕√n, degrees of freedom (df = n − 1), critical value t*, rejection region, one-sided vs. two-sided alternative, paired data, mean difference μd, test–interval duality, conditions for inference
Materials slides (Deck 14), the Week 14 chapter (with the course t-table), the week's readings + video links, a spreadsheet (Google Sheets or Excel), a Desmos-class graphing/statistics tool, 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 (callback to Week 13). "Last week we ran the courtroom: H₀ is presumed innocent, evidence is measured with a p-value, and the verdict vocabulary is reject or fail to reject. But every p-value last week was handed to you. Today the training wheels come off: you compute the evidence." Put a promise on the screen: "Average delivery: 50 minutes." — "You time 25 deliveries yourself and get 52. Is that a broken promise, or just sampling noise? By the end of today you can settle it with arithmetic."

  • The tool is the one-sample t-test — the same t machinery as Week 11's confidence intervals, pointed at a claim instead of an estimate.
  • "This is the hinge week of the inference half: Week 13 gave the logic, this week gives the machinery for means, next week aims the same machinery at proportions and at two groups."

The promise (write it on the board): "By the end of this week you can take any claimed average — a delivery promise, an advertised mpg, a 'before vs. after' — and run the full trial: hypotheses, evidence, verdict, in plain English."

Why it matters line (memory hook): "A t-statistic counts standard errors between the data and the claim."

Segment 2 — The One-Sample t-Test, Start to Finish (24 min)

Plain language first. Week 13's recipe never changes; we're just filling in the middle step. The whole test is five moves:

  1. State the claim as hypotheses — H₀: μ = μ₀ (the promised value), Hₐ: μ ≠ μ₀ (or < or >, chosen before the data — Segment 3).
  2. Check the conditions (Segment 7 details them).
  3. Measure the distance in standard errors — that's the test statistic:

t = (x̄ − μ₀) ⁄ SE, where SE = s ⁄ √n — "sample mean minus claimed mean, divided by the standard error."

  1. Get the cutoff — the critical value t* from the course t-table at df = n − 1 (or a p-value from technology).
  2. Verdict, in context — |t| beyond t* → reject H₀; otherwise fail to reject. Say it in words about the claim, never just "reject."

Memory hook: "Claim → ruler → distance → cutoff → verdict." The ruler is SE — Week 10 built it (σ∕√n), Week 11 taught us to use s in its place, and that swap is exactly why the cutoff comes from the t-table, not the z-table.

THE COURSE t-TABLE (identical to Week 11's — the only critical values this week uses):

The table below gives critical values t* by degrees of freedom and confidence level.

df t* for 90% t* for 95% t* for 99%
9 1.833 2.262 3.250
15 1.753 2.131 2.947
24 1.711 2.064 2.797

(Reading it for tests: a two-sided test at α = 0.05 uses the 95% column; at α = 0.01, the 99% column. A one-sided test at α = 0.05 uses the 90% column — its value leaves exactly 5% in one tail. Any other df: technology supplies the value.)

One fully worked example (do every step out loud).

A courier service advertises: "Average delivery: 50 minutes." A city consumer-affairs office times a random sample of 25 deliveries: x̄ = 52 minutes, s = 5 minutes. Test the claim at α = 0.05.
- Hypotheses: H₀: μ = 50 vs. Hₐ: μ ≠ 50 (the office is checking the promise, not gunning for one direction — two-sided).
- SE = 5 ⁄ √25 = 5 ⁄ 5 = 1 minute.
- t = (52 − 50) ⁄ 1 = 2.0 — the sample sits 2 standard errors above the promise.
- Cutoff: df = 25 − 1 = 24; two-sided at α = 0.05 → t* = 2.064.
- Verdict: |2.0| < 2.064 → fail to reject H₀. Technology sharpens it: p ≈ 0.057, just above 0.05. In words: "the sample does not provide convincing evidence that the true average delivery time differs from 50 minutes."

Land the two subtleties while the example is warm:
- The verdict was close — 2.0 vs. 2.064. Statistics is not vibes; the cutoff decides, and honest analysts report the p-value (0.057) so readers see how close.
- We did not prove the promise true. More on that in Segment 4 — it's the week's #1 trap.

Segment 3 — One Tail or Two? Choosing the Alternative (20 min)

Plain language first. The alternative hypothesis is the question, and it must be chosen before the data arrive:

  • Two-sided (Hₐ: μ ≠ μ₀): "is the promise wrong, in either direction?" — the default for checking a claim.
  • One-sided (Hₐ: μ < μ₀ or μ > μ₀): "we suspect a specific direction" — legitimate only when the direction was the question from the start (a watchdog suspects less mpg; a training program should raise scores).

The mechanics differ in one place: a one-sided test puts all of α in one tail, so its cutoff is smaller — at α = 0.05 that's the 90% column of the table (1.833 / 1.753 / 1.711) — and its p-value is half the two-sided p (when the data lean the suspected way).

One fully worked example (the flip — do every step).

A compact car model advertises 40 mpg highway. A consumer magazine — which suspects the real figure is lower — runs 16 standardized test drives: x̄ = 38.4 mpg, s = 3.2.
- SE = 3.2 ⁄ √16 = 3.2 ⁄ 4 = 0.8. t = (38.4 − 40) ⁄ 0.8 = −2.0. df = 15.
- One-sided (Hₐ: μ < 40, the magazine's stated suspicion): cutoff −1.753 → −2.0 is beyond it → reject H₀ (technology: p ≈ 0.032). "Convincing evidence the true average is below 40 mpg."
- Two-sided (Hₐ: μ ≠ 40): cutoff ±2.131 → |−2.0| < 2.131 → fail to reject (p ≈ 0.064).
Same data, different question, different verdict. That's not a scandal — it's why the question must be locked in before the data are seen.

Misconception + cure:
- ❌ Peeking at the data, noticing x̄ < μ₀, and then "choosing" the one-sided test that rejects.
✅ Cure: the alternative comes from the question, not the numbers. Choosing the tail after peeking secretly doubles your Type I error rate. Memory hook: "Pick your tail before you peek." If in doubt, go two-sided — it's the honest default.

Segment 4 — Misconceptions + Think-Pair-Share (23 min) · Session 1 closes (~75)

Name the misconceptions out loud, then cure each:

  • ❌ "We failed to reject, so the average really is 50 — claim confirmed."
    ✅ Cure: not guilty ≠ innocent. Failing to reject means the evidence didn't clear the bar, nothing more. Our courier sample (p ≈ 0.057) was a whisker from rejecting — calling that "confirmed" is spin. (Week 13's line, now with numbers attached.)
  • ❌ Dividing by s instead of SE. t = (52 − 50)/5 = 0.4 — "looks fine!"
    ✅ Cure: the ruler for sample means is s∕√n (Week 10's engine). Dividing by s alone shrinks every t by √n and nothing ever rejects. Compute SE first, on its own line, every time.
  • ❌ Using z* = 1.96 (or the Week 8 z-table) with a small sample.
    ✅ Cure: with s standing in for σ, small samples need the wider t cutoffs — 2.064 at df 24, 2.262 at df 9 — precisely Week 11's lesson. If your cutoff for n = 10 is 1.96, you're borrowing certainty you don't have.
  • ❌ "t = 2.0 means the mean is twice the claim" (or "2 minutes off").
    ✅ Cure: t counts standard errors, not minutes and not multiples. Say it in words: "2 standard errors above the claimed mean."
  • ❌ Statistical significance = practical importance (and its mirror).
    ✅ Cure: with a huge n, a trivial 0.2-minute gap can reject; with a small n, a real 2-minute gap can fail to reject (ours did!). Report the size of the effect alongside the verdict, always.

Interaction — Think-Pair-Share (rapid-fire, ~10 min). Six quick items on a slide; solo 30 sec, pair 1 min, fingers vote:
1. A test uses n = 25. df? → 24
2. s = 5, n = 25. SE? → 1
3. t = 2.5, two-sided α = 0.05, df 24. Verdict? → reject (2.5 > 2.064)
4. t = −1.2, cutoff ±2.262. Verdict? → fail to reject (|−1.2| < 2.262)
5. Each of 12 delivery vans is driven one week on route plan A and one week on plan B. Paired or independent? → paired (same van, twice)
6. A one-sided p is 0.03. The two-sided p for the same t? → 0.06

Segment 5 — Paired Data: Subtract First (25 min) · Session 2 opens

Hook back in: "A typing course promises to make an office faster. You measure everyone before and after. Two columns of numbers — but not two separate groups. The same ten people appear in both. That's paired data, and it gets its own move."

Plain language first.
- Paired data = two measurements on the same individual (before/after, left/right, with/without) — or on deliberately matched pairs.
- The move: compute each pair's difference d, then run an ordinary one-sample t-test on the differences, testing H₀: μd = 0 ("no change on average"). Memory hook: "Pairs? Subtract first — then it's one sample."
- Why subtract? Each person is their own control. Fast typists stay fast, slow stay slow — subtracting cancels that person-to-person spread, leaving only the change. Ignoring the pairing buries the signal in noise.

One fully worked example (do every step out loud).

An office of 10 staff members takes a two-week typing course. Words-per-minute, before and after:

The table below lists each staffer's speeds and the difference d = after − before.

Staffer 1 2 3 4 5 6 7 8 9 10
Before 38 45 36 50 41 44 39 47 42 48
After 47 44 44 50 47 46 43 51 46 52
d 9 −1 8 0 6 2 4 4 4 4
  • d̄ = (9 − 1 + 8 + 0 + 6 + 2 + 4 + 4 + 4 + 4) ⁄ 10 = 40 ⁄ 10 = 4 wpm.
  • sd: the squared deviations from 4 sum to 90, so sd² = 90 ⁄ 9 = 10 and sd = √10 ≈ 3.162.
  • SE = 3.162 ⁄ √10 = 1 exactly (√10 ⁄ √10). t = (4 − 0) ⁄ 1 = 4.0. df = 10 − 1 = 9.
  • Verdict: two-sided at α = 0.05, t* = 2.262 → 4.0 is far beyond → reject H₀ (technology: p ≈ 0.003). "Convincing evidence the course changed average typing speed — and the change is an improvement, about 4 wpm."

Misconception + cure:
- ❌ Treating the two columns as two independent groups (or comparing only the two column averages with no test at all).
✅ Cure: the pairing is information — person 4's 50-before explains their 50-after. One question decides the analysis: "Is each number in column one linked to one specific number in column two?" Linked → paired → subtract first. (Genuinely independent groups arrive next week.)

Segment 6 — Tests ↔ Intervals: Two Tools, One Answer (18 min)

Plain language first. Week 11's confidence interval and this week's two-sided test are the same machine running in opposite directions:

  • The 95% CI lists every value of μ the data find plausible.
  • The two-sided test at α = 0.05 asks whether one particular value (μ₀) is on that list.

The duality: μ₀ inside the 95% CI ⟺ fail to reject H₀: μ = μ₀ at α = 0.05 (two-sided). Outside ⟺ reject. Same s, same √n, same t* — it has to agree.

Worked example (close the loop with the courier).

The courier sample: x̄ = 52, SE = 1, df = 24, t* = 2.064. The 95% CI is 52 ± 2.064(1) = (49.94, 54.06) minutes (unrounded: 49.936 to 54.064).
- Is 50 on the list? Yes → the two-sided test must fail to reject. It did.
- Now read what the interval adds: every mean from ~49.9 to ~54.1 minutes is plausible. The promise survives, but so does "54 minutes" — which is exactly why "fail to reject" is not "confirmed." The interval shows how wide the reasonable doubt is.

Memory hook: "The interval lists every claim that would survive; the test tries one." Best practice in real reports: give both — the verdict and the interval.

Misconception + cure:
- ❌ "The test said fail-to-reject, so the CI must be wrong when it includes values far from μ₀."
✅ Cure: they can't disagree (two-sided, matching α) — the CI is the set of all values a test wouldn't reject. If the interval looks embarrassingly wide, that's not a contradiction; that's the test's weakness made visible. Small n → wide interval → weak test.

Segment 7 — The Fine Print: Conditions (12 min)

Plain language first. The t-test inherits Week 11's conditions, word for word:

  1. Random & representative — the sample was collected by a chance method from the population the claim is about (Week 1 never leaves).
  2. Independent observations — one measurement doesn't influence another.
  3. Roughly normal population, or a large enough n — the t procedures tolerate mild non-normality, and the CLT (Week 10) covers larger samples; but with small n (like our n = 10 and 16), a strongly skewed histogram or a wild outlier is a deal-breaker. For paired data, the conditions apply to the differences.

The practical habit: before any small-sample t-test, make a quick dot plot or histogram of the data (or the differences). One extreme value in ten observations can single-handedly manufacture — or destroy — a rejection. If the picture looks wrong, say so in the report; don't run the test silently.

Misconception + cure:
- ❌ "The software printed a p-value, so the test is valid."
✅ Cure: software checks nothing. =T.TEST will happily analyze a sample of receipts your cousin picked because they were on top. Conditions are the analyst's job — the printout can't see the study design.

Segment 8 — Technology Workflow + AI-Critique, Callback & Hand-off (20 min) · Session 2 closes (~75)

Technology workflow (exact steps, live in a spreadsheet):
1. Data in A2:A26 (the 25 delivery times). x̄: =AVERAGE(A2:A26) → 52. s: =STDEV(A2:A26) → 5. SE: =STDEV(A2:A26)/SQRT(25) → 1. t: =(52-50)/1 → 2.0 — build it in steps, one cell each, so every piece is checkable.
2. p-value (two-sided): =T.DIST.2T(2, 24) → 0.0569 — feed it |t| and df. One-sided: =T.DIST(-2, 15, TRUE) → 0.0320 (area in the left tail, for the mpg example).
3. The table's source: =T.INV.2T(0.05, 24) → 2.064 — the course t-table is this function, three rows at a time.
4. Paired in one shot: with Before in B2:B11 and After in C2:C11, =T.TEST(B2:B11, C2:C11, 2, 1) → 0.003 (tails = 2, type = 1 means paired). It matches our hand-built t = 4.0 — the point of doing one by hand first.
5. Desmos-class tool: tdist(24) draws the t curve; shade beyond ±2 and read the two-sided area live.

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

Paste this to your chatbot: "A sample of 25 has mean 52 and standard deviation 5. Test the claim that the population mean is 50 at the 5% level — show the t-statistic and your critical value, no tools."
Then check it against this week's machinery: the correct t is (52 − 50)/(5/√25) = 2.0. Chatbots routinely divide by s alone (t = 0.4), grab 1.96 from the normal table instead of 2.064 from t at df 24 (which flips the verdict!), or round the p-value into a confident "reject." The tool drafts, you judge — and this week the stakes are a verdict, not just a number.

Callback + tease:
- Callback: "Week 10 built the SE ruler, Week 11 turned it into intervals, Week 13 wrote the courtroom rules — this week they merged into a working trial for any claimed average."
- Tease next week: "Two things left to test: percentages (does the claimed 40% renewal rate survive?) and the question every experiment really asks — are these two groups different from each other? Same recipe, new fuel."

Hand-off (the week's work):
- Chapter 14 (primary reading, with the course t-table) — then Lecture Tutorial 14 (AI tutor; share link + summary).
- Data Lab 14 ("Put the Field Guide on Trial" — your own penguin sample vs. a published reference value) · Quiz 14 (end of week) · Discussion 14 ("The 'On Average' Alibi") · Assignment 14 (AI-coached).

Instructor FAQ — Common Stumbles

Student says / does Quick cure
"We failed to reject — so the claim is true, right?" Not guilty ≠ innocent. The evidence didn't clear the bar; that's all. Show the courier CI (49.94–54.06): "54 minutes is also still plausible — did we prove that too?"
Computes t = (52 − 50)/5 = 0.4. The ruler for a mean is SE = s∕√n, not s. Make SE its own line in every solution — the √n is where the sample size earns its keep.
Uses 1.96 as the cutoff for n = 10. Small samples pay a t premium: df 9 → 2.262. z* sneaks back in constantly; the table row must match df = n − 1.
Picks the one-sided test after seeing x̄ < μ₀. "Pick your tail before you peek." The alternative is the question, not the result. Default to two-sided when unsure.
Runs two-independent-style reasoning on before/after columns. Ask: "is each number linked to one partner in the other column?" Linked → paired → subtract first, then one-sample t on d.
"p = 0.057 means there's a 5.7% chance the claim is true." Week 13's cure, again: p is the probability of data this extreme if the claim is true. It judges the data, not the hypothesis.
"t = 2.0 but the table says 2.064 — can't we call it close enough?" The cutoff is the agreed rule; report "fail to reject" AND the p (0.057) so the reader sees the closeness honestly. Changing the rule after the fact is how p-hacking starts.
"My df is 21 — it's not in the table." By design: the course table carries df 9, 15, 24. Off-table df → technology (=T.INV.2T(0.05, 21)); the table teaches the pattern, software handles the rest.
"The difference is significant, so it matters." Significance ≠ size. A 0.2-minute delay can be "significant" at n = 10,000 and still be nothing. Always report the effect in real units next to the verdict.

Scope flag

This outline stays within Objective 7's means portion. The test ↔ interval duality segment (reinforcing Objective 6), the =T.TEST/=T.DIST.2T technology walk, and the paired-vs-independent recognition drill are added context (not strictly required by the objective) — kept because they cure the week's misconceptions and set up Week 15; cut them for a leaner session. Two-sample procedures are deliberately deferred to Week 15.