Week 8 — Lecture Outline · Midterm Review & Exam

Course: Introduction to Statistics (MATH 11) · Silver Oak University (fictional sample) · Prof. Rivera
Objectives covered: cumulative — Objectives 1–4 (Weeks 1–7). Obj 1 — populations vs. samples, sampling & design; Obj 2 — summarize & display univariate data (shape, center, spread); Obj 3 — relationships between two variables; Obj 4 — probability rules, conditional probability, random variables (incl. binomial & normal models).
SLOs touched: A (reason quantitatively from data) · B (communicate results to a non-technical audience)
Meeting pattern: 2 sessions × 75 min = 150 min. Segment minutes below total ~150; scale to your own pattern.

This is a review-and-exam week — no new content. Each segment briskly re-teaches one theme from Weeks 1–7 with one worked example and the single misconception most likely to cost points, then the final segment frames the midterm itself. Built to be taught from cold as a review: an instructor (or a substitute) can run it without having taught the first seven weeks, because every definition and cure travels with the segment. The midterm covers Objectives 1–4; it does not reach the normal/sampling-distribution and inference material that begins in Week 9.

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Week at a Glance

The week's big question	"Across the whole first half — getting data, describing it, relating it, and reasoning about chance — what is the one honest move each topic asks of us, and where does everyone slip?"
By the end of the week, students can…	(1) re-derive each objective's core move on demand — classify a variable and a sampling method (Obj 1); summarize a dataset's shape/center/spread and pick the honest number (Obj 2); read a scatterplot/correlation without sliding into causation (Obj 3); apply the probability rules, conditional probability, and expected value, and run a binomial or normal-approximation calculation (Obj 4); (2) name and avoid the highest-cost misconception in each theme; (3) walk into the Midterm knowing its format, its weight (20%), and a concrete preparation plan built around the Study Guide, the Exam-Prep Tutorial, and the Practice Exam.
Key vocabulary (all review)	population/sample, parameter/statistic, NOIR (nominal/ordinal/interval/ratio), SRS / stratified / cluster / systematic, bias, observational vs. experiment, confounding; histogram, shape/skew, outlier, mean/median/mode, variance & SD, five-number summary, IQR, resistant measure; scatterplot, correlation r, lurking variable; sample space, probability rules (complement, addition, multiplication), conditional probability P(A|B), independence, random variable, expected value E(X), the binomial setting & B(n, p), normal model & the 68–95–99.7 rule, normal approximation to the binomial
Materials	slides (Deck 8 — the review deck), the Study Guide, the Exam-Prep Tutorial (AI), the Practice Exam, a spreadsheet (Google Sheets or Excel), one approved chatbot (Gemini / Claude / ChatGPT) for the audit-the-AI review moment
Timing note	8 segments, ~150 min total. Session 1 (Tue) = Segments 1–4 (~75): Objectives 1–2. Session 2 (Thu) = Segments 5–8 (~75): Objectives 3–4 + the midterm frame. Scale to your own pattern.

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Segment 1 — Hook & the Map of the First Half (8 min) · Session 1 opens

Hook. Put one number on the board with no label: 0.62. Ask: "Is this a parameter or a statistic? Is it trustworthy? Is it even the right kind of number for what someone's about to claim?" Let them realize they can't answer until they ask where it came from and what it describes.
- "That instinct — to interrogate a number before believing it — is the entire first half of this course. Today we walk the whole arc once, fast, and find the exact spot in each topic where points get lost."

The promise (write it on the board): "By Thursday you'll be able to take any of the four big skills — get data, describe it, relate it, reason about chance — and on demand state the one honest move it requires and the one mistake that sinks it. That's the midterm."

The map (one slide, say it out loud — this is the photograph slide of the week):

Obj 1 — GET the data (who, how, what kind). Obj 2 — DESCRIBE one variable (shape, center, spread). Obj 3 — RELATE two variables (scatter, correlation — not cause). Obj 4 — REASON about chance (probability rules, conditional, expected value, binomial & normal).

Why it matters line (memory hook): "Weeks 1–7 are one sentence: get good data, describe it honestly, relate it carefully, and reason about it with the rules of chance."

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Segment 2 — Objective 1 Review: Where Data Comes From (16 min)

Re-teach in plain language. A population is everyone the question is about; a sample is the part we measured. A number describing the population is a parameter (p, μ); the matching sample number is a statistic (p̂, x̄). "The hat means measured, not true." Then two more questions decide trust: how were they picked (SRS / stratified / cluster / systematic = trustworthy; convenience / voluntary response = traps), and what kind of variable was recorded (NOIR — nominal, ordinal, interval, ratio).

One worked example (do every step out loud):

Claim: "62% of Silver Oak undergrads feel stressed about money." A pollster emails all 18,000 undergrads; 900 reply; 558 say yes.
- 558/900 = 0.62 → 62%: a statistic (from the sample). The true campus-wide figure is the parameter p, which we never see directly.
- "Feels stressed: yes/no" is nominal; if they'd recorded a 1–5 stress rating it would be ordinal; dollars of debt would be ratio.
- The 900 who replied are a voluntary-response slice of the 18,000 — watch for nonresponse bias (the most-stressed may be likeliest to answer). Method, not size, decides trust.

Highest-cost misconception + cure:
- ❌ "It's a number, so it's quantitative," and "a bigger sample is automatically better."
✅ Cure: a zip code or jersey number is a nominal label — you can't average it. And size never fixes bias: Literary Digest 1936 had 2.4 million responses and called the wrong winner. Method beats size.

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Segment 3 — Objective 2 Review: Describing One Variable (24 min)

Re-teach in plain language. Once you have good data, you describe one variable three ways: its shape (from a histogram — symmetric, or skewed left/right, and any outliers), its center (one typical number), and its spread (how tightly it clusters).
- Center: mean (the balance point — chases outliers), median (the middle — resistant), mode (most frequent — the only center for categorical data).
- Spread: standard deviation (typical distance from the mean — pairs with the mean), and the IQR = Q3 − Q1 from the five-number summary (min, Q1, median, Q3, max — pairs with the median, resistant).
- The pairing rule: symmetric, no big outliers → report mean + SD. Skewed or outliers → report median + IQR.

One worked example (compute it live):

Quiz scores: 2, 4, 4, 5, 5, 5, 6, 7, 9, 53 (the 53 is a data-entry typo for 5-point-something — leave it in to make the point).
- Mean = 100/10 = 10.0. Median = average of the 5th and 6th values = (5+5)/2 = 5.0.
- The lone 53 drags the mean up to 10 while the median sits calmly at 5. Under that outlier, median + IQR tells the truth; mean + SD lies.
- Five-number summary: min 2, Q1 = 4, median 5, Q3 = 7, max 53 → IQR = 7 − 4 = 3. A box-and-whisker would flag 53 as a clear outlier.

Highest-cost misconception + cure:
- ❌ "Always report the average," and "a bigger SD means the data is wrong."
✅ Cure: under skew or outliers the mean is the dishonest one — switch to the median. And SD just measures spread; large SD = more variable, not "more error." "The mean chases the outlier; the median ignores it."

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Segment 4 — Objective 2, the Computation Reflex + Quick Drill (12 min) · Session 1 closes (~75)

The reflex to lock in (say it twice). On the exam you will not be asked to derive a variance from scratch under time pressure; you'll be asked to choose and interpret. So the reflex is: glance at the shape first, then pick the matching pair. Skew or an outlier in the picture → median + IQR. Roughly symmetric → mean + SD.

Interaction — rapid-fire "which pair?" (think-pair-share, ~8 min):
Put four one-line scenarios on a slide; students decide mean+SD or median+IQR, solo (30 s), neighbor (1 min), then vote by fingers (1 = mean+SD, 2 = median+IQR).

1. Heights of students in this room. (symmetric → mean+SD)
2. Household incomes in a city. (right-skewed → median+IQR)
3. Time to finish a 10-question quiz, one student fell asleep. (outlier → median+IQR)
4. Scores on an easy quiz where most got 9–10. (left-skewed → median+IQR)

Debrief the one that splits the room (#1 vs. the rest): the only "mean+SD" case is the one with no skew and no outlier.

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Segment 5 — Objective 3 Review: Relating Two Variables (18 min) · Session 2 opens

Hook back in: "Session 1 we described one variable at a time. Now: two variables at once — and the single most expensive mistake in statistics lives here."

Re-teach in plain language. For two quantitative variables, a scatterplot shows the relationship; the correlation r measures the strength and direction of the straight-line part.
- r runs from −1 to +1. Sign = direction (positive = move together, negative = opposite). Magnitude = strength (near ±1 = tight line; near 0 = no linear pattern). r = 0 means no linear relationship — a perfect U-shape can still have r ≈ 0.
- For two categorical variables you use a two-way table instead.
- The headline: a strong r is a link, not a push. Only a randomized experiment can claim cause; an observational correlation can always hide a lurking (confounding) variable.

One worked example (read it, don't compute it):

Headline: "Students who drink more coffee get higher grades — r = 0.6." Observational.
- r = 0.6: a moderate positive linear link. It does not say coffee raises grades.
- Lurking variable: hours studying could drive both the coffee and the grades. Nothing was randomly assigned, so the arrow is unproven.
- Honest report (SLO B): "Coffee and grades tend to rise together here, but we can't say coffee causes the rise — a third factor like study time may explain both."

Highest-cost misconception + cure:
- ❌ "Strong correlation proves X causes Y," and "r = 0 means no relationship at all."
✅ Cure: ask "was anything randomly assigned?" — if no, it's a link. And r only sees the straight-line part; a clear curved pattern can have r ≈ 0. "Correlation is a handshake, not a push."

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Segment 6 — Objective 4 Review (Part 1): The Rules of Chance (20 min)

Re-teach in plain language. Probability is the language of "how surprised should I be?" Four moves carry the whole unit:
- Basic probability: P(event) = favorable ÷ total when outcomes are equally likely; always between 0 and 1.
- Complement: P(not A) = 1 − P(A). (The fastest tool for "at least one.")
- Addition (OR): P(A or B) = P(A) + P(B) − P(A and B); subtract the overlap so you don't double-count. (For mutually exclusive events the overlap is 0.)
- Conditional & multiplication: P(A and B) = P(A) · P(B | A). Events are independent when P(B | A) = P(B) — knowing A tells you nothing about B.

One worked example (do every step):

Draw one card from a standard 52-card deck.
- P(king) = 4/52 = 1/13. P(not a king) = 1 − 1/13 = 12/13 (complement).
- P(king or heart) = 4/52 + 13/52 − 1/52 = 16/52 = 4/13 (subtract the king-of-hearts overlap once).
- P(second card is a king given the first was a king, no replacement) = 3/51 — the condition changed the sample space, so the draws are not independent.

Highest-cost misconception + cure:
- ❌ "Just add the two probabilities," and "P(A and B) = P(A)·P(B) always."
✅ Cure: for OR, subtract the overlap unless the events truly can't co-occur. For AND, multiply only if independent — otherwise use the conditional P(B | A). When stuck on "at least one," reach for the complement first.

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Segment 7 — Objective 4 Review (Part 2): Random Variables, Binomial & Normal (22 min)

Re-teach in plain language. A random variable attaches a number to a chance outcome (heads in 10 flips; cars through a light). Discrete = countable list of values; continuous = any value in a range. The expected value E(X) is the long-run average: E(X) = Σ [value × its probability] — a weighted average, not the most likely single value.

Two working models to recognize:
- Binomial — B(n, p): the setting is fixed n trials, two outcomes each, constant p, independent trials (mnemonic: BINS). Then E(X) = n·p and SD = √(n·p·(1−p)). Use it for "how many successes out of n?"
- Normal model: the bell curve for continuous data, summarized by the 68–95–99.7 rule — about 68% within 1 SD of the mean, 95% within 2, 99.7% within 3. When n is large, a binomial count is approximately normal with the same mean n·p and SD √(n·p·(1−p)).

One worked example (compute it live):

A free-throw shooter makes 70% of shots; she takes n = 10, shots independent. Let X = makes. This is B(10, 0.7) (fixed n, make/miss, constant p, independent → BINS all check).
- E(X) = n·p = 10 × 0.7 = 7 makes. SD = √(10 × 0.7 × 0.3) = √2.1 ≈ 1.45.
- "About 7, give or take ~1.5" is the one-line read. By the 68–95–99.7 intuition, roughly 7 ± 1.45 covers the typical range.
- Quick complement tie-back: P(misses all 10) = 0.3¹⁰ ≈ 0.0000059 — so P(at least one make) ≈ 0.9999941.

Highest-cost misconception + cure:
- ❌ "Expected value is the most likely value," and "any 'how many' problem is binomial."
✅ Cure: E(X) is a long-run average and can be a value X never actually takes (E = 2.5 makes is impossible on any single trial). And binomial needs all of BINS — fixed n, two outcomes, constant p, independent. Drawing without replacement breaks "constant p / independent."

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Segment 8 — The Midterm Frame: What's On It & How to Prepare (12 min) · Session 2 closes (~75)

Audit-the-AI review moment (the course's recurring habit, one last time before the exam):

Paste to an approved chatbot: "In a sample survey, is a 5-point satisfaction rating interval or ratio? And does a correlation of 0.8 mean the cause is established?"
Check it against what we taught. Chatbots often call a rating "interval" (it's ordinal — the gaps aren't equal) and may soften the correlation/causation line. The tool drafts; you judge. If you can catch the model here, you're ready.

What's on the Midterm (state it plainly — put it on the closing slide):
- Coverage: cumulative over Weeks 1–7, Objectives 1–4 — getting data, describing one variable, relating two variables, and probability/random variables (incl. binomial & normal models). It does not include the normal-distribution calculations, sampling distributions, or inference that start in Week 9.
- Weight & logistics: the Midterm is 20% of the course grade. The window opens Mon Oct 19 and the exam is due Sun Oct 25, 11:59 p.m. (There is no regular Quiz 8 or Assignment 8 — the midterm replaces them.)
- Format: a mix of classify/compute/interpret items in the spirit of the worked examples above — choosing the honest summary, reading a correlation, applying a probability rule, recognizing a binomial setting and finding E(X).

The preparation plan (point at each artifact by name):
1. Study Guide — work it first; it's the checklist of every move in the four objectives.
2. Exam-Prep Tutorial — run it with an approved chatbot (Gemini / Claude / ChatGPT) and submit the share link; it drills the weak spots adaptively.
3. Practice Exam — sit it timed, like the real thing, then review what you missed against the Study Guide.
4. Discussion 8 (the debrief) — after the exam, reflect on one idea from the first half that changed how you read a real number in the world.

Callback + tease:
- Callback: "Every item on Thursday's exam is a move you already made in Weeks 1–7 — today we just named it and found where it slips."
- Tease next: "After the midterm, Week 9 turns the bell curve into a precision instrument — z-scores and exact normal probabilities — and from there the whole back half: sampling distributions, confidence intervals, and hypothesis tests."

Hand-off (the week's work): review the Study Guide, run the Exam-Prep Tutorial (share link), take the Practice Exam, sit the Midterm (due Sun Oct 25), and post Discussion 8 (the debrief, due Sun Oct 25).

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Instructor FAQ — Common Stumbles (Review Week)

Student says / does	Quick cure
"Which do I report — mean or median?"	Look at the shape first. Skew or an outlier → median + IQR. Roughly symmetric, no outlier → mean + SD. The picture decides.
Calls a strong correlation "proof" of cause.	Ask: was anything randomly assigned? No → it's a link, not a cause. Hunt the lurking variable.
"r = 0, so the two variables are unrelated."	r only measures the straight-line part. A clear U-shape can have r ≈ 0 and still be a strong (curved) relationship.
Adds two probabilities for an "OR" and gets > 1.	Use P(A or B) = P(A) + P(B) − P(A and B) — subtract the overlap. Only skip the overlap if the events are mutually exclusive.
Multiplies P(A)·P(B) for "AND" without checking.	Multiply only if independent. Otherwise P(A and B) = P(A)·P(B | A). Sampling without replacement breaks independence.
"Expected value is the most likely outcome."	E(X) is the long-run average (Σ value × probability). It can be a number X never actually equals — E(X) = 2.5 is fine.
"Is this binomial?" — unsure.	Run BINS: Binary outcome, Independent trials, fixed N trials, Same probability p. All four → binomial, with E = n·p, SD = √(n·p(1−p)).
Treats a 1–5 rating as ratio/interval.	A rating is ordinal — ordered, but "4 vs. 5" isn't a measured, equal gap and there's no true zero. Don't average it as if it were ratio.
Panics that the exam is "everything."	It's Objectives 1–4 only (Weeks 1–7). The normal/inference machinery (Weeks 9+) is not on the midterm. Bound the studying.

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Scope flag

This outline is pure review of Objectives 1–4 — no new material. The few framing extras (the unlabeled-0.62 cold open, the Literary Digest and Bezos callbacks, the BINS mnemonic) are retained context carried over from Weeks 1–7 because they make the cures stick; cut them for a leaner 60-minute review. The midterm and its bundle (Study Guide, Exam-Prep Tutorial, Practice Exam) are built separately and only referenced here by name.