Week 16 · Study guide

Final Exam Study Guide · Weeks 1–15 (Objectives 1–8)

Introduction to Statistics · MATH 11 Fall 2026 · Prof. Rivera Fictional sample

Course: Introduction to Statistics (MATH 11) · Silver Oak University (fictional sample) · Prof. Rivera
This is a student-facing review page. Read it, work the fresh practice, and follow the dated plan. Then run the paired Exam-Prep Tutorial and take the Practice Final for active recall. (This guide points to those two — it does not repeat them.)

Integrity note for students. Every practice item on this page is a fresh variant — new numbers and contexts — with a pre-computed, vetted answer. None of these are the live final questions. Working them builds the skill the final tests, the honest way.

What the final covers (read this first)


Exam	Final — cumulative, Weeks 1–15, all 8 Objectives
Format	25 items, 100 points. Application-skewed: most items ask you to do something with data or interpret a result, not just recite a definition. Expect multiple-choice, short numeric answers, and several "read this output and explain" items.
Coverage (where the points are)	Obj 1 ≈ 2 items (sampling & study design) · Obj 2 ≈ 3 (summarizing data, center/spread) · Obj 3 ≈ 2 (relationships, correlation, two-way tables) · Obj 4 ≈ 4 (probability, random variables, binomial & normal models) · Obj 5 ≈ 3 (the normal distribution + sampling distributions/CLT) · Obj 6 ≈ 4 (confidence intervals — t for means, z for proportions) · Obj 7 ≈ 4 (hypothesis testing + tests for means/proportions) · Obj 8 ≈ 3 (linear regression & inference for the slope). The inference half — Objectives 6, 7, 8 — is the heaviest single block (~11 of 25). Budget the most time there.
Weight	The final is 30% of your course grade — the single biggest assessment in the course.
When it opens / where	Opens in the Week 16 module (the final-review-and-exam week). The exam window and the room/timing are posted with the exam itself in Canvas; this guide and the exam-prep tutorial post before it so you can prepare. There is no weekly quiz, assignment, or discussion in Week 16 — the final replaces them.
What to bring	A calculator and the one-page formula list you build from this guide. *The z-table, the t values, and the z* values are supplied on the exam — every problem is engineered to land on the friendly values you used all term; you never recall or look one up.** Spreadsheets are for practice at home; the exam is do-it-and-interpret.

How to use this guide. Each objective below has the same four parts: (A) the key ideas in plain language, (B) the definitions / formulas / procedures, (C) the predictable mistakes and their cures, and (D) where to review in the module. After all eight objectives come fresh worked examples + self-check questions (with answers), a dated study plan sized to finals week, and how it's graded + test strategy.

The supplied tables (the same ones from class — they appear on the exam).

z-table (cumulative area to the LEFT of z). Every normal/CLT problem lands on one of these.

z −3.0 −2.5 −2.0 −1.5 −1.0 −0.5 0 +0.5 +1.0 +1.5 +2.0 +2.5 +3.0

area to LEFT .0013 .0062 .0228 .0668 .1587 .3085 .5000 .6915 .8413 .9332 .9772 .9938 .9987

Three moves: left = read it; right = 1 − left; between = (left of bigger) − (left of smaller).

t* critical values (confidence intervals for a mean; df = n − 1).

Confidence df 9 (n=10) df 15 (n=16) df 24 (n=25) df ∞ (= z)

90% 1.833 — — 1.645

95% 2.262 2.131 2.064 1.960

z* critical values (proportions, and large-sample tests). 90% → 1.645 · 95% → 1.960 · 99% → 2.576.
Test critical values: one-sided z at α=0.05 → 1.645; two-sided z at α=0.05 → ±1.96; one-sided t at α=0.05, df 24 → 1.711.

z	−3.0	−2.5	−2.0	−1.5	−1.0	−0.5	0	+0.5	+1.0	+1.5	+2.0	+2.5	+3.0
area to LEFT	.0013	.0062	.0228	.0668	.1587	.3085	.5000	.6915	.8413	.9332	.9772	.9938	.9987

Confidence	df 9 (n=10)	df 15 (n=16)	df 24 (n=25)	df ∞ (= z)
90%	1.833	—	—	1.645
95%	2.262	2.131	2.064	1.960

Objective 1 — Foundations & Study Design (Week 1) · ~2 items

(A) Key ideas, plain language

Statistics is about trusting a number that describes people you didn't count. Three questions decide whether a statistic deserves your trust: Who was measured? How were they picked? What was recorded? A population is everyone the question is about; a sample is the part you actually measured. The whole course is the bridge from a statistic (what you have) to a parameter (what you want).

(B) Definitions, formulas, procedures

Population = everyone/everything the question is about. Sample = the part measured. Census = measuring the whole population.
Parameter = a number describing the population. Statistic = a number from the sample. Memory hook: P*opulation → Parameter, Sample → Statistic.* Notation: population proportion p, mean μ; sample proportion p̂, mean x̄. The hat means "measured," not "true."
Levels of measurement — NOIR: Nominal (names, no order — major, zip code, jersey number), Ordinal (ordered, unequal gaps — letter grade, S/M/L, class standing), Interval (ordered, equal gaps, no true zero — °F, °C, calendar year), Ratio (ordered, equal gaps, true zero — height, age, income, counts). The test: Does zero mean "none"? yes → ratio; equal gaps, arbitrary zero? → interval; ordered labels, fuzzy gaps? → ordinal; just names? → nominal.
Sampling methods: SRS (every individual equally likely — gold standard) · Stratified (split into groups, random-sample within each) · Cluster (randomly pick whole groups, measure everyone) · Systematic (every k-th after a random start). The bad ones: convenience and voluntary response (usually biased).
Bias = error baked into the method, pushing results the same wrong way no matter the sample size (undercoverage, nonresponse, response, voluntary-response).
Observational study = watch and record. Experiment = impose a treatment and compare; only a randomized experiment supports a cause-and-effect claim. Confounding variable = a third variable tangled with both.

(C) Predictable mistakes → cures

❌ "If it's a number, it's quantitative." → ✅ Zip codes and jersey numbers label. Ask does arithmetic mean anything? — nominal.
❌ "A bigger sample is automatically better." → ✅ Size never fixes bias. The 1936 Literary Digest poll got 2.4 million replies and still called the wrong winner. Method beats size.
❌ "Population means a lot of people." → ✅ It's whoever the question is about — could be 25 students. A role, not a size.
❌ Mixes stratified and cluster. → ✅ Stratified = sample within every group; cluster = sample whole groups.
❌ Strong correlation → cause. → ✅ Ask was anything randomly assigned? If no, it's a link, not a cause; hunt the confounder.

(D) Review in the module

Week 1 → Lecture Outline (Segments 2–7), Slides (Deck 1), Readings, Lecture Tutorial 1.

Objective 2 — Summarizing One Variable (Weeks 2–3) · ~3 items

(A) Key ideas, plain language

First see the data (a table, then a histogram); then summarize it with one number for the center and one for the spread — and match those numbers to the shape. Under skew or outliers, the mean and SD lie, and the median and IQR tell the truth. Shape decides the summary.

(B) Definitions, formulas, procedures

Frequency = a count. Relative frequency = frequency ÷ total (sums to 1). Histogram = picture of quantitative data; bars touch. Bar chart = categorical; bars have gaps.
Shapes: symmetric, skewed right (long tail to big values), skewed left (long tail to small values), uniform, bimodal. Skew is named for the tail. Describe in order — S-C-S-O: Shape, Center, Spread, Outliers.
Mean x̄ = (Σx) ÷ n. Median = middle of sorted values (even count → average the two middles). Mode = most frequent (only center for categorical data).
Variance & SD (sample): deviation = value − mean; square the deviations; variance s² = (sum of squared deviations) ÷ (n − 1); SD s = √variance. (Whole population → ÷ n; a sample → ÷ (n − 1).)
Five-number summary: Min · Q1 · Median · Q3 · Max. Q1 = median of the lower half; Q3 = median of the upper half. IQR = Q3 − Q1. Range = Max − Min.
Resistance: median & IQR are resistant; mean & SD are non-resistant. Pair them: mean ↔ SD, median ↔ IQR.
Skew-vs-center: right-skew → mean > median; left-skew → mean < median; symmetric → about equal.

(C) Predictable mistakes → cures

❌ "Skewed left means it leans left." → ✅ Skew is named for the tail: long thin tail to the left (small values), tall bulk on the right.
❌ Adds deviations and expects a nonzero spread. → ✅ Plain deviations always sum to 0 — that's why we square them.
❌ Divides the sample variance by n. → ✅ Sample variance/SD divides by n − 1.
❌ Reports the mean with the IQR (or median with SD). → ✅ Keep the couple together: mean ↔ SD, median ↔ IQR.
❌ Reports the mean for obviously skewed/outlier data. → ✅ The mean chases the outlier; report the median under skew.

(D) Review in the module

Week 2 → Lecture Outline (frequency tables, histograms, shape), Slides (Deck 2), Lecture Tutorial 2. Week 3 → Lecture Outline (mean/median/mode, variance & SD, five-number summary, resistance), Slides (Deck 3), Lecture Tutorial 3.

Objective 3 — Relationships Between Two Variables (Week 4) · ~2 items

(A) Key ideas, plain language

When two things move together, you can picture it (a scatterplot), measure it (correlation r), or tabulate it (a two-way table) — and you still cannot draw the causal arrow without an experiment or a ruled-out lurking variable. Correlation is a handshake, not a push.

(B) Definitions, formulas, procedures

Scatterplot: two quantitative variables, one dot per individual. Explanatory (x) explains; response (y) responds. Describe with D-F-S: Direction (positive/negative), Form (linear/curved/none), Strength (tight/moderate/loose) — plus any outliers?
Correlation r: one number for the direction and strength of a linear relationship. Always between −1 and +1. r = +1 perfect up line; r = −1 perfect down; r = 0 no linear relationship. Rough guide: |r| 0.8–1.0 strong, 0.5–0.8 moderate, < 0.3 weak. r is unitless, symmetric, not a percent.
Two-way (contingency) table: cross-classifies two categorical variables. Marginal proportion = margin total ÷ grand total. Conditional proportion = restrict to one group first, then ÷ that group's total (the word "given" picks the denominator). Conditional proportions compare groups.
Lurking / confounding variable = a third variable that drives both and manufactures a misleading association. Correlation ≠ causation. Two-question cure: (1) Could a third variable explain both? (2) Was anything randomly assigned?

(C) Predictable mistakes → cures

❌ "r = 0 means the variables aren't related." → ✅ Only the linear part is 0. A clean U-curve can have r ≈ 0 and still be tightly related. Picture before number.
❌ "Stronger r means a steeper line." → ✅ Strength = how tight the dots hug the line; slope = how steep. r measures scatter, not slope.
❌ Divides a two-way cell by the grand total when the question says "of the exercisers…". → ✅ "Given"/"of" names the group → divide by that group's total. Restrict first.
❌ "r = 0.6 means 60%." → ✅ r is not a percent. Report sign + size in words.
❌ Sees strong r and says "X causes Y." → ✅ Name a plausible lurking variable; ask whether anything was randomized.

(D) Review in the module

Week 4 → Lecture Outline (scatterplots D-F-S, correlation r, two-way tables, lurking variables), Slides (Deck 4), Lecture Tutorial 4.

Objective 4 — Probability & Random Variables (Weeks 5–7) · ~4 items

(A) Key ideas, plain language

Put an honest number on what you don't yet know, and learn how new information changes it. Then turn outcomes into numbers (random variables) and ask: what should I expect on average (E(X)), and how much will it swing (σ)? Finally, the binomial — counts of yes/no successes — and when the bell curve can stand in for it.

(B) Definitions, formulas, procedures

Probability rules:
- Sample space S = the complete list of outcomes; event = a subset. Equally-likely: P(event) = favorable ÷ total. Every probability is in [0, 1]; all outcomes sum to 1.
- Complement: P(not A) = 1 − P(A). "At least one" → 1 − "none."
- Addition (OR): P(A or B) = P(A) + P(B) − P(A and B). Mutually exclusive → overlap 0.
- Multiplication (AND): if independent, P(A and B) = P(A) × P(B).
- Conditional: P(A | B) = P(A and B) ÷ P(B) — of the times B happened, how often did A? On a two-way table, "given" shrinks the world to one row/column. Independence test: P(A | B) = P(A). Mutually exclusive ≠ independent. P(A | B) ≠ P(B | A) in general (base-rate trap).

Random variables:
- X attaches a number to a chance outcome. Discrete = countable; continuous = fills an interval. Valid distribution: every P in [0, 1] and ΣP = 1.
- Expected value E(X) = Σ [ x · P(X=x) ] — the weighted average.
- Variance Var(X) = E(X²) − [E(X)]², where E(X²) = Σ [ x² · P(X=x) ]; σ = √Var(X). Mean of the squares minus the square of the mean.
- Bets/insurance: judge by E(net gain); negative E favors the other side.

Binomial & normal model:
- BINS (all four): Binary outcome, Independent trials, N fixed number of trials, Same p. ("Without replacement" or "until I get one" breaks it.)
- Binomial probability: P(X = k) = C(n, k) · pᵏ · (1−p)ⁿ⁻ᵏ. Choose values: n=2 → 1,2,1; n=3 → 1,3,3,1; n=4 → 1,4,6,4,1.
- Binomial mean & SD: μ = np, σ = √(np(1−p)).
- Normal-approximation license: use it only when np ≥ 10 AND n(1−p) ≥ 10.

(C) Predictable mistakes → cures

❌ "After a streak, the other outcome is 'due.'" → ✅ Independent trials have no memory (gambler's fallacy).
❌ "For 'A or B,' just add." → ✅ Only if mutually exclusive; otherwise subtract the overlap (King-or-Heart = 16/52).
❌ Swaps P(A|B) and P(B|A). → ✅ The thing after the bar is the world you're in; with a rare condition the base rate dominates.
❌ Treats any table as a distribution. → ✅ Must pass both gates: each P in [0,1] and ΣP = 1.
❌ Reports E(X²) as the variance. → ✅ Subtract [E(X)]². There's no n to divide by.
❌ Uses √(np) for the SD. → ✅ SD = √(np(1−p)) — the (1−p) factor is essential.
❌ Applies the normal curve to small n. → ✅ Check np ≥ 10 and n(1−p) ≥ 10 first.

(D) Review in the module

Week 5 → Lecture Outline (probability rules, conditional), Deck 5, Tutorial 5. Week 6 → Lecture Outline (random variables, E(X), variance), Deck 6, Tutorial 6. Week 7 → Lecture Outline (BINS, binomial, normal-approx check), Deck 7, Tutorial 7.

Objective 5 — The Normal Distribution & Sampling Distributions (Weeks 9–10) · ~3 items

(A) Key ideas, plain language

A z-score turns any value into one honest sentence: how many standard deviations from average am I, and is that unusual? The bell curve's area is a proportion, so a z + a table answers any "what share is above/below/between?" question. Then the twist: even when the data aren't normal, the averages of samples are — the Central Limit Theorem — and that bell, centered at μ with spread σ/√n, is the doorway to all of inference.

(B) Definitions, formulas, procedures

The normal distribution (Week 9):
- A density curve has total area 1, so any area under it is a proportion. A normal curve N(μ, σ) is symmetric and bell-shaped; the mean splits it in half.
- 68–95–99.7 (empirical) rule: about 68% within 1σ, 95% within 2σ, 99.7% within 3σ. Tails: outside ±1σ → 16% each; outside ±2σ → 2.5% each (the working definition of "unusual").
- z-score: z = (value − mean) ÷ SD = (x − μ) ÷ σ. Positive = above the mean, negative = below. Three table moves: area left = read it; right = 1 − left; between = (left of bigger) − (left of smaller).
- Reverse: a percentile → its z → un-standardize: value = μ + z·σ.
- Assessing normality: the empirical rule and z-scores are trustworthy only when the data are roughly bell-shaped (symmetric, single-peaked, no wild outliers). Skewed data (income, home prices) → don't use them.

Sampling distributions (Week 10):
- A statistic like x̄ is a moving target — sampling variability. The sampling distribution of x̄ has center = μ and spread = the standard error = σ/√n (tighter than individuals; tighter still as n grows). Quadruple n to halve the SE.
- Central Limit Theorem (CLT): for large n (rule of thumb n ≥ 30), x̄ is approximately normal whatever the population's shape. Then x̄ ~ N(μ, σ/√n), and you standardize with σ/√n: z = (x̄ − μ) ÷ (σ/√n).

(C) Predictable mistakes → cures

❌ "A negative z is an error." → ✅ Negative just means below the mean (z = −2 is two SDs below — valid and unusual-on-the-low-side).
❌ Reports the left area when the question asks "above." → ✅ "Above" = right = 1 − left. Draw the curve and shade the asked-for side.
❌ Applies 68–95–99.7 to skewed data. → ✅ First test normality; if skewed, the rule doesn't apply.
❌ Uses σ as the spread of the average. (The signature CLT error.) → ✅ The spread of an average is σ/√n, not σ. Always divide by √n for a mean.
❌ "The CLT makes the data normal." → ✅ It makes the averages normal; a skewed population stays skewed.
❌ Forgets the √ and uses σ/n. → ✅ It's σ over √n (σ = 60, n = 9 → 60/3 = 20).

(D) Review in the module

Week 9 → Lecture Outline (density curves, empirical rule, z-scores, areas/percentiles, assessing normality), Deck 9, Tutorial 9. Week 10 → Lecture Outline (sampling variability, SE = σ/√n, the CLT), Deck 10, Tutorial 10.

Objective 6 — Confidence Intervals (Weeks 11–12) · ~4 items

(A) Key ideas, plain language

From one sample, state an honest range for the unknown truth — a mean (with t) or a proportion (with z) — and attach a confidence level without overclaiming. Every interval is estimate ± margin of error: a center (the best guess) and a reach (how far you extend each side). The skill students lose the most points on is saying what "95% confident" means — and refusing to say what it doesn't.

(B) Definitions, formulas, procedures

CI for a mean (Week 11):
- Use t (not z) because σ is unknown and you estimate it with the sample's s; the t-distribution has fatter tails, indexed by df = n − 1.
- CI: x̄ ± t* · (s / √n). SE = s/√n. Margin of error ME = t* · SE. Four steps: SE → t* (supplied, df = n−1) → ME → x̄ ± ME.
- As df grows, t* shrinks toward the z value (1.960 at 95%).

CI for a proportion (Week 12):
- Use z* — no degrees of freedom for proportions.
- CI: p̂ ± z* · √( p̂(1−p̂) / n ). SE = √(p̂(1−p̂)/n). ME = z* · SE.
- Sample size for a target margin: n = (z*/ME)² · p̂(1−p̂), using p̂ = 0.5 for the conservative worst case (maximizes p̂(1−p̂) = 0.25). Always round n UP.
- Conditions: random; independent (sample < 10% of population); large counts for proportions: np̂ ≥ 10 AND n(1−p̂) ≥ 10.

What moves the margin: bigger n → smaller ME (quadruple n to halve it); higher confidence → bigger t*/z* → bigger ME; for proportions, p̂ near 0.5 → bigger ME.

The one correct interpretation: "We are 95% confident the true population mean/proportion lies between ___ and ___" — shorthand for "a method that captures the truth 95% of the time in the long run."

(C) Predictable mistakes → cures

❌ Uses 1.96 for a small sample with unknown σ. → ✅ 1.96 is z; with unknown σ you owe the wider t (e.g., 2.064 at df 24). Using z makes the interval too narrow.
❌ Uses t and df for a proportion. → ✅ Proportions use z* — no df.
❌ "95% of the data fall in the interval." → ✅ A CI brackets the mean/proportion, not individual values.
❌ "There's a 95% chance the true value is in THIS interval." → ✅ Once computed, the interval either captures the truth or not; the 95% describes the method over many samples.
❌ Forgets to divide by √n (means) or rounds sample size DOWN (proportions). → ✅ SE uses √n; sample size always rounds up (600.25 → 601).
❌ Thinks 99% is "more accurate." → ✅ Higher confidence → wider (less precise) interval.

(D) Review in the module

Week 11 → Lecture Outline (t vs z, building a CI, margin of error, the two interpretation traps), Deck 11, Tutorial 11. Week 12 → Lecture Outline (proportion CI, choosing a sample size, conditions), Deck 12, Tutorial 12.

Objective 7 — Hypothesis Testing (Weeks 13–14) · ~4 items

(A) Key ideas, plain language

When a sample shows an effect, decide whether it's real or just the wobble random chance produces all the time. A test is a courtroom: H₀ = innocent (no effect / chance), Hₐ = guilty (a real effect), the data are the evidence, and α is "beyond a reasonable doubt." Every test runs the same four beats — State → Compute → Compare → Conclude — and the conclusion is a sentence about the world, not just "reject H₀."

(B) Definitions, formulas, procedures

The logic (Week 13):
- H₀ always contains an equals idea (μ = 75, p = 0.5); Hₐ is ≠, >, or <. Hypotheses are about the population parameter, never the sample. The thing you hope to show goes in Hₐ.
- p-value = the probability of data at least as extreme as observed, assuming H₀ is true — how surprising the data would be if nothing were going on. Decision rule: p ≤ α → reject H₀; p > α → fail to reject H₀. Default α = 0.05.
- Type I error = reject a true H₀ (false positive; its probability is α). Type II error = fail to reject a false H₀ (false negative). "Type I cries wolf; Type II misses the wolf."
- Statistical vs. practical significance: significant = probably real, not big/important. A huge sample can make a trivial effect significant.

The mechanics (Week 14) — State → Compute → Compare → Conclude:
- One-sample t-test (a mean): t = (x̄ − μ₀) / (s/√n), df = n − 1.
- One-proportion z-test: z = (p̂ − p₀) / √( p₀(1−p₀)/n ) — the SE uses p₀ (the null value), because everything is computed assuming H₀ is true. Enter proportions as decimals.
- Two-sample comparison: interpret only — H₀: the two means are equal; read the supplied p-value, compare to α, conclude.
- Choosing the test: (1) mean (average/measurement) → t; proportion (share/rate/%) → z. (2) one group vs. a number → one-sample; two groups vs. each other → two-sample. Count the groups.
- One- vs. two-sided: "above/more/less" → one-sided; "different/changed" → two-sided.

(C) Predictable mistakes → cures

❌ Puts the claim to prove in H₀. → ✅ H₀ is the claim you try to knock down; the exciting claim is Hₐ.
❌ Writes hypotheses about x̄ / p̂. → ✅ Hypotheses are about μ / p (the unknown truth); the sample is the evidence.
❌ "p = 0.03 means a 3% chance H₀ is true." → ✅ p is computed assuming H₀ is true, so it can't measure the null.
❌ "Fail to reject" = "H₀ proven true." → ✅ "Not guilty" ≠ "innocent." We never accept H₀.
❌ "Significant = large/important." → ✅ Significant = probably real; ask for the effect size.
❌ Drops the √n in the t denominator, or uses p̂ in a proportion test SE. → ✅ Denominator is s/√n; a proportion test uses p₀ (a CI uses p̂).
❌ Reports only "reject H₀." → ✅ Finish the sentence: "At the 0.05 level, we have significant evidence that [the real-world claim]."

(D) Review in the module

Week 13 → Lecture Outline (H₀/Hₐ, p-value vs α, Type I/II, the three misreads), Deck 13, Tutorial 13. Week 14 → Lecture Outline (one-sample t, one-proportion z, two-sample idea, choosing the test), Deck 14, Tutorial 14.

Objective 8 — Linear Regression & Inference for the Slope (Week 15) · ~3 items

(A) Key ideas, plain language

Once two things move together, draw the line that turns the pattern into a prediction — then the grown-up move: ask whether the line is real or just noise. Correlation says they move together; regression draws the line that lets you bet on the next one; inference tells you whether the line is worth betting on. A line still isn't a cause.

(B) Definitions, formulas, procedures

Least-squares line: ŷ = b₀ + b₁x. ŷ ("y-hat") = predicted y. b₁ = slope = change in ŷ per one-unit increase in x (carries units). b₀ = intercept = predicted ŷ when x = 0.
Interpret in context, with units: "for each additional hour, predicted score rises about 4 points"; "a student at 0 hours is predicted to score 50" (check whether x = 0 is sensible).
Prediction: plug x into the line. Residual = observed y − predicted ŷ (always observed minus predicted). Positive → above the line; negative → below.
r² (coefficient of determination) = the share of the variation in y the line explains = correlation squared; between 0 and 1 (report as a percent). Slope = how much; r² = how well.
Inference for the slope: H₀: slope = 0 (no linear relationship); Hₐ: slope ≠ 0. A t-test for the slope hands you a t and a p-value; compare to α. p < α → slope is significantly different from 0 (real relationship); p ≥ α → could be noise.
Residual plot: a patternless cloud around 0 → a straight line fits. A curve (U/arch) → the relationship was curved (wrong model). Fanning out → non-constant spread.
Extrapolation: the line is trustworthy only inside the data's x-range. Predicting outside it is where regression lies.

(C) Predictable mistakes → cures

❌ "The slope is 4" (no units/context). → ✅ Say it in words: 4 points of score per additional hour studied.
❌ Computes the residual as predicted − observed. → ✅ It's observed − predicted, every time. Sign carries meaning.
❌ "r² is the slope." → ✅ Slope = per-unit change (units); r² = unitless share of variation (0.81 = 81%). Steep ≠ high r².
❌ Plugs a far-out x into the line. → ✅ Extrapolation. Stay in the data's range (40 hours → 210 on a 100-pt exam is nonsense).
❌ "A significant slope (or high r²) proves x causes y." → ✅ Inference tests slope ≠ 0, never causation. Observational → a link, not a cause.
❌ Reads a U-shaped residual plot as "fine." → ✅ A pattern means a straight line is the wrong model, however small the residuals.

(D) Review in the module

Week 15 → Lecture Outline (least-squares line, slope/intercept in context, prediction & residuals, r², inference for the slope, residual plots), Deck 15, Tutorial 15.

Representative practice (all fresh — vetted answers)

None of these are live final items. New numbers, new contexts. Each answer is pre-computed; the one-line why names the idea it tests. Cover the answers, work each one, then check. Practice is weighted toward the heavier objectives (4–8).

Objective 1 practice

Worked example — population/sample, statistic/parameter, NOIR, sampling method.
A streaming service emails all 40,000 subscribers and 1,000 reply; 620 rate a new feature "useful." It also records each respondent's satisfaction tier (Bronze/Silver/Gold) and hours watched last week (a count).
- (a) Population? Sample? (b) Is 620/1,000 a statistic or a parameter, as a percent? (c) Classify satisfaction tier and hours watched. (d) If they had instead split subscribers by country and randomly sampled within each country, name the method.
Answer. (a) Population = all 40,000 subscribers; sample = the 1,000 who replied. (b) Statistic: 620 ÷ 1,000 = 0.62 = 62%. (c) Satisfaction tier = ordinal (ordered, unequal gaps); hours watched = ratio (a count, true zero). (d) Stratified. Why: a sample number is a statistic estimating the unseen parameter; NOIR is settled by order and whether zero means "none"; sampling within every group = stratified.

Self-check (Obj 1).
1. True/false: a census is a kind of sample. → False — a census measures the whole population.
2. A study finds gym members sleep better (nothing assigned). Cause or link? → Link (observational; a lurker like overall health habits could drive both).
3. Classify jersey number. → Nominal (a numeric label).
4. A 2-million-reply opt-in web poll vs. a 1,000-person SRS — which is more trustworthy? → The SRS (method beats size).

Objective 2 practice

Worked example 1 — mean/median/SD + which to trust.
Six repair costs ($): 80, 95, 100, 110, 120, 415.
- (a) Mean and median. (b) Which center is honest here, and why?
Answer. (a) Mean = (80+95+100+110+120+415) ÷ 6 = 920 ÷ 6 ≈ $153.33; median = average of 3rd and 4th = (100+110)/2 = $105. (b) The median ($105) — the lone $415 is an outlier dragging the mean above five of the six values. Why: mean & SD are non-resistant; under an outlier, report the median (and IQR).

Worked example 2 — sample SD from scratch.
Five values: 3, 5, 5, 7, 10.
- Find the mean, then the sample variance and SD.
Answer. Mean = (3+5+5+7+10) ÷ 5 = 30 ÷ 5 = 6. Deviations: −3, −1, −1, +1, +4 (sum 0 ✓); squares: 9, 1, 1, 1, 16; sum = 28; variance = 28 ÷ (5−1) = 7; SD = √7 ≈ 2.65. Why: square the deviations (they otherwise sum to 0); sample divides by n − 1.

Worked example 3 — five-number summary + IQR.
Seven values (sorted): 12, 15, 18, 20, 24, 27, 40.
- Give the five-number summary, the IQR, and the range.
Answer. Min = 12, Max = 40, Median (4th) = 20. Lower half {12,15,18} → Q1 = 15; upper half {24,27,40} → Q3 = 27. Summary = 12 · 15 · 20 · 27 · 40; IQR = 27 − 15 = 12; range = 40 − 12 = 28. Why: Q1/Q3 are the medians of the halves; IQR ignores the outlier that inflates the range.

Self-check (Obj 2).
1. Histogram or bar chart for blood type? → Bar chart (categorical).
2. Mean 64, median 50 — likely shape? → Skewed right (mean > median).
3. You report the median. Which spread pairs with it? → IQR.
4. Switching a histogram to relative frequency changes the shape? → No (only relabels the vertical axis).

Objective 3 practice

Worked example — two-way table: conditional + marginal + independence.
250 commuters answered Carpool? and On time?

	On time	Late	Row total
Carpools	72	8	80
Drives alone	119	51	170
Column total	191	59	250

(a) P(on time given carpools). (b) P(on time given drives alone). (c) Marginal P(carpools). (d) Are carpooling and on time independent?
Answer. (a) 72 ÷ 80 = 0.90 = 90%. (b) 119 ÷ 170 = 0.70 = 70%. (c) 80 ÷ 250 = 0.32 = 32%. (d) No — P(on time | carpools) = 0.90 ≠ P(on time) = 191/250 = 0.764, so dependent. Why: "given" picks the denominator; a gap (90% vs 70%) is association, not causation; independence is a number check.

Self-check (Obj 3).
1. r for a perfect U-shaped scatter? → ≈ 0 (no linear relationship).
2. If you swap x and y, does r change? → No (r is symmetric).
3. "Towns with more churches have more bars." Lurking variable? → Town population/size.
4. r = 0.7 — does that mean "70% related"? → No (r is not a percent).

Objective 4 practice — largest pre-inference section; work all of these

Worked example 1 — addition, complement, conditional from a setup.
Draw one card from a standard 52-card deck.
- (a) P(King or Heart). (b) P(not a face card). (Face cards J, Q, K = 12.) (c) P(Heart | red card). (26 red cards; 13 hearts.)
Answer. (a) 4/52 + 13/52 − 1/52 = 16/52 = 4/13 ≈ 0.308. (b) 1 − 12/52 = 40/52 = 10/13 ≈ 0.769. (c) 13 ÷ 26 = 0.5. Why: "or" subtracts the overlap; complement = 1 − P; "given red" shrinks the world to 26 cards.

Worked example 2 — independence (multiplication & "at least one").
A fair coin is flipped and a fair die is rolled.
- (a) P(tails and a 5). (b) P(at least one 5 in two die rolls).
Answer. (a) (1/2)(1/6) = 1/12 ≈ 0.083. (b) 1 − (5/6)(5/6) = 1 − 25/36 = 11/36 ≈ 0.306. Why: "and" with independence multiplies; "at least one" = 1 − "none."

Worked example 3 — expected value & SD of a random variable.
X takes values 0, 1, 2, 3 with probabilities 0.4, 0.3, 0.2, 0.1.
- (a) Confirm valid. (b) E(X). (c) Var(X) and σ.
Answer. (a) Each in [0,1] and 0.4+0.3+0.2+0.1 = 1.00 ✓. (b) E(X) = 0(0.4)+1(0.3)+2(0.2)+3(0.1) = 0+0.3+0.4+0.3 = 1.0. (c) E(X²) = 0+1(0.3)+4(0.2)+9(0.1) = 0.3+0.8+0.9 = 2.0; Var = 2.0 − (1.0)² = 1.0; σ = √1.0 = 1.0. Why: E(X) is the weighted average; Var = E(X²) − [E(X)]².

Worked example 4 — expected value of a bet.
A game costs $4 to play. With probability 0.10 you win $30; otherwise $0.
- Find the expected net value. Good deal?
Answer. Net: win = $30 − $4 = +$26 (p = 0.10); lose = −$4 (p = 0.90). E(net) = 26(0.10) + (−4)(0.90) = 2.60 − 3.60 = −$1.00. Bad deal — you lose about $1 per play on average. Why: judge a bet by E(net); negative favors the house.

Worked example 5 — binomial probability + mean/SD + normal-approx check.
An email link is clicked by each of n = 200 independent recipients with probability p = 0.1. X = number of clicks.
- (a) P(exactly 2 clicks among the first 3 recipients). (b) μ and σ for all 200. (c) Is the normal approximation licensed?
Answer. (a) For 3 recipients, X ~ B(3, 0.1): C(3,2)·(0.1)²·(0.9)¹ = 3 · 0.01 · 0.9 = 0.027. (b) μ = np = 200 × 0.1 = 20; σ = √(np(1−p)) = √(200 × 0.1 × 0.9) = √18 ≈ 4.24. (c) np = 20 ≥ 10 ✓ and n(1−p) = 180 ≥ 10 ✓ → yes. Why: P(X=k)=C(n,k)pᵏ(1−p)ⁿ⁻ᵏ; μ = np, σ = √(np(1−p)); normal approx needs both np and n(1−p) ≥ 10.

Self-check (Obj 4).
1. Is "deal 4 cards, count aces" binomial? → No (without replacement → not independent, p changes).
2. A distribution lists P = 0.35, ?, 0.20, 0.15. The missing probability? → 1 − (0.35+0.20+0.15) = 0.30.
3. For X ~ B(50, 0.1), is the normal approximation allowed? → np = 5 < 10 → No.
4. A test is "97% accurate" for a disease 1% have, and you test positive — chance you're sick ≈ 97%? → No (base rate makes it far lower).

Objective 5 practice

Worked example 1 — z-score, percentile, and "between" area.
Battery lifetimes are roughly normal with μ = 500 hours, σ = 50 hours.
- (a) Standardize a 550-hour battery and give its percentile. (b) What share last between 450 and 600 hours? (c) What lifetime sits at the 97.72nd percentile?
Answer. (a) z = (550 − 500)/50 = +1.0 → left area .8413 → 84th percentile. (b) 450 → z = −1.0; 600 → z = +2.0; between = .9772 − .1587 = .8185 ≈ 81.85%. (c) left area .9772 → z = +2.0 → value = μ + z·σ = 500 + 2(50) = 600 hours. Why: standardize, then the three table moves; reverse a percentile with value = μ + z·σ.

Worked example 2 — empirical rule.
Heights are roughly normal with μ = 68 in, σ = 3 in.
- (a) What % are between 65 and 71 inches? (b) What % are above 74 inches?
Answer. (a) 65 and 71 are μ ± 1σ → about 68%. (b) 74 = μ + 2σ → upper tail beyond +2σ ≈ 2.5%. Why: 68–95–99.7; outside ±2σ is 5%, split into 2.5% per tail.

Worked example 3 — standard error + CLT probability.
Daily purchase amounts are right-skewed with μ = $4.00, σ = $3.00. A manager samples n = 36 purchases.
- (a) Is x̄ approximately normal? Why? (b) Standard error. (c) P(x̄ < $3.50)?
Answer. (a) Yes — n = 36 ≥ 30, so by the CLT x̄ is approximately normal even though purchases are skewed. (b) SE = σ/√n = 3.00/√36 = 3.00/6 = $0.50. (c) z = (3.50 − 4.00)/0.50 = −1.0 → left area .1587 → about 15.87%. Why: the CLT bells the averages; standardize a sample mean with σ/√n, not σ.

Self-check (Obj 5).
1. A z-score of −2.5 means? → 2.5 SDs below the mean (unusual, low side).
2. Population SD σ = 60; for n = 100 the standard error is? → 60/√100 = 6.
3. Does a bigger sample shrink σ? → No — it shrinks the standard error σ/√n, not σ.
4. "Above" a value asks for the left or right area? → Right = 1 − left.

Objective 6 practice

Worked example 1 — confidence interval for a mean (t).
A random sample of n = 25 study sessions has mean length x̄ = 45 min, sample SD s = 15 min. Build a 95% CI for the true mean.
- Steps and interval.
Answer. SE = s/√n = 15/√25 = 15/5 = 3. df = 24 → t* (95%) = 2.064 (supplied). ME = 2.064 × 3 = 6.192 ≈ 6.19. Interval = 45 ± 6.19 = (38.81, 51.19) → report (38.8, 51.2). Why: CI = x̄ ± t*·(s/√n); unknown σ → t with df = n − 1.

Worked example 2 — confidence interval for a proportion (z).
A poll of n = 400 voters finds p̂ = 0.55 support a measure. Build a 95% CI.
- Steps and interval; check large counts.
Answer. SE = √(0.55 × 0.45 / 400) = √(0.2475/400) = √0.00061875 ≈ 0.0249. z* (95%) = 1.960 (supplied). ME = 1.960 × 0.0249 ≈ 0.0488 ≈ 0.049. Interval = 0.55 ± 0.049 = (0.501, 0.599) → about 50.1% to 59.9%. Large counts: np̂ = 220 ≥ 10, n(1−p̂) = 180 ≥ 10 ✓. Why: CI = p̂ ± z*·√(p̂(1−p̂)/n); proportions use z*, no df.

Worked example 3 — choosing a sample size (worst case).
A pollster wants a 95% interval with a ±0.05 (5-point) margin and no prior estimate.
- Required sample size?
Answer. Use p̂ = 0.5. (z*/ME)² = (1.960/0.05)² = (39.2)² = 1536.64; × 0.25 = 384.16; round UP → n = 385. Why: n = (z*/ME)²·p̂(1−p̂), worst case p̂ = 0.5 maximizes p̂(1−p̂) = 0.25; always round up.

Self-check (Obj 6).
1. "95% of the data fall in the interval" — right? → No (a CI brackets the mean/proportion, not individuals).
2. For n = 16, unknown σ, 95% — multiplier 1.96 or 2.131? → 2.131 (t at df 15).
3. Does a 99% CI get narrower or wider than a 95%? → Wider.
4. Sample size came out 600.2 — report 600 or 601? → 601 (round up).

Objective 7 practice

Worked example 1 — one-sample t-test (full pipeline).
A coach claims a drill raises players' average free-throw makes (out of 20) above the team norm of 12. A sample of n = 25 gives x̄ = 13, s = 5. Technology supplies p ≈ 0.16. Test at α = 0.05.
- State → Compute → Compare → Conclude.
Answer. H₀: μ = 12; Hₐ: μ > 12. SE = 5/√25 = 1; t = (13 − 12)/1 = 1.00 (df 24). Compare: p ≈ 0.16 > 0.05 → fail to reject H₀. Conclude: "At the 0.05 level, we do not have significant evidence that the drill raises the average above 12." (Not "the drill doesn't work.") Why: t = (x̄−μ₀)/(s/√n); p > α → fail to reject; conclude in context.

Worked example 2 — one-proportion z-test (full pipeline).
A site claims more than 40% of visitors subscribe. A sample of n = 100 finds p̂ = 0.50. Technology supplies p ≈ 0.021. Test at α = 0.05.
- State → Compute → Compare → Conclude.
Answer. H₀: p = 0.40; Hₐ: p > 0.40. SE = √(0.40 × 0.60 / 100) = √(0.24/100) = √0.0024 ≈ 0.049. z = (0.50 − 0.40)/0.049 ≈ 2.04. Compare: p ≈ 0.021 ≤ 0.05 → reject H₀. Conclude: "At the 0.05 level, there is significant evidence that more than 40% of visitors subscribe." Why: a proportion test SE uses p₀; reject when p ≤ α; finish with a sentence about the world.

Worked example 3 — choosing the test + Type I/II.
For each, name the test (and one- vs. two-sided): (a) "Is the mean checkout time different from 90 seconds?" (one sample of times). (b) "Do more than 25% of orders use a coupon?" (one sample, a rate). (c) "Does Version A's mean revenue differ from Version B's?" (d) In (a), describe a Type I error.
Answer. (a) One-sample t, two-sided ("different"). (b) One-proportion z, one-sided ("more than"). (c) Two-sample comparison (two groups). (d) Type I = rejecting a true H₀ = concluding the mean differs from 90 sec when it really doesn't (a false alarm). Why: mean → t, proportion → z; one group vs. a number → one-sample, two groups → two-sample; Type I = false positive.

Self-check (Obj 7).
1. "p = 0.04 means a 4% chance the null is true" — right? → No (p assumes H₀ true; it can't measure the null).
2. p = 0.20, α = 0.05 — reject or fail to reject? → Fail to reject.
3. A statistically significant result must be a large effect? → No (significant = probably real, not big).
4. Which value goes in the proportion test's SE — p̂ or p₀? → p₀ (the null value).

Objective 8 practice

Worked example 1 — read a regression line: slope, intercept, prediction, residual.
A regression of monthly sales y ($1,000s) on ad spend x ($1,000s) gives ŷ = 8 + 3x, fit on spends of 1–10.
- (a) Interpret the slope and intercept in context. (b) Predict sales for x = 6. (c) A store spent 6 and actually sold $25k; find the residual. (d) Is predicting for x = 50 sensible?
Answer. (a) Slope 3: each extra $1,000 in ad spend predicts about $3,000 more in sales; intercept 8: a store with $0 ad spend is predicted to sell about $8,000 (x = 0 is at/near the edge — flag). (b) ŷ = 8 + 3(6) = $26,000. (c) residual = observed − predicted = 25 − 26 = −$1,000 (below the line). (d) No — x = 50 is far outside the 1–10 data → extrapolation. Why: slope = per-unit change with units; residual = observed − predicted; the line only holds inside the data's range.

Worked example 2 — r² and inference for the slope.
The same regression reports r² = 0.64, t = 5.1, p = 0.0003. Use α = 0.05.
- (a) Interpret r². (b) Is the slope statistically significant? (c) Does this prove ad spend causes higher sales?
Answer. (a) 64% of the variation in sales is explained by ad spend (the rest is other factors + scatter). (b) Yes — p = 0.0003 < 0.05 → reject H₀ (slope = 0); there is real evidence of a linear relationship. (c) No — significance and a high r² describe and predict, but this is observational; a lurker (e.g., store size or season) could drive both. Why: r² = share of variation explained; p < α → slope ≠ 0; correlation ≠ causation.

Self-check (Obj 8).
1. Residual = predicted − observed? → No — it's observed − predicted.
2. r² = 0.81 — what does it say? → 81% of the variation in y is explained by x.
3. p = 0.30 for the slope, α = 0.05 — real relationship or noise? → Could be noise (fail to reject H₀).
4. A U-shaped residual plot means? → A straight line is the wrong model (the relationship is curved).

Study plan — a dated countdown (finals week, sized to 2 sessions/week)

Built for the Week 16 final. Adjust the exact dates to your section's posted exam day; the rhythm is what matters. The final is cumulative and the inference half (Obj 6–8) is the heaviest — start there once your foundations are warm. Do a little every day rather than one long cram.

When	Do this (≈60–90 min)
~7 days out (end of Week 15)	Read this guide's Objectives 1–3 (the describe-and-relate foundation). Work the Obj 1, 2, 3 practice. Build your one-page formula sheet (NOIR test, S-C-S-O, mean/SD, five-number summary, r, two-way conditional rule).
~6 days out	Read Objective 4 carefully and work all of its worked examples (probability, E(X)/σ, binomial, normal-approx check). Re-derive any you missed.
~5 days out	Read Objective 5 (z-scores, the three table moves, the empirical rule, the CLT with σ/√n). Work its practice. Add the z-table moves and SE = σ/√n to your formula sheet.
~4 days out	Read Objective 6 (CIs: t for means, z for proportions, sample size, conditions). Work its practice. Memorize the one correct interpretation of "95% confident" and the two traps.
~3 days out	Read Objectives 7 & 8 (the State → Compute → Compare → Conclude machine; choosing the test; the slope test, r², residuals, extrapolation). Work their practice. Then run the paired Exam-Prep Tutorial (`N-exam-prep-tutorial-week-16`) in an approved chatbot — it diagnoses your weak spots across all 8 objectives and drills them.
~2 days out	Take the Practice Final (`O-practice-final-week-16`, the paired practice exam in this module) under timed, closed-note conditions. Score it; list every missed idea by objective.
~1 day out	Re-teach only the topics you missed on the practice final (use this guide's mistake-cures and the relevant Lecture Tutorial). Re-do those specific self-checks, with extra attention to Obj 6–8. Sleep.
Exam day	Skim your one-page formula sheet. Arrive early. Read each item twice; for every inference item, name the four beats (State → Compute → Compare → Conclude).

Two paired tools — use both (don't skip):
- Exam-Prep Tutorial (N-exam-prep-tutorial-week-16) — a copy/paste chatbot tutor that diagnoses, re-teaches, and drills you across all 8 objectives, ending with a readiness summary. Best for active recall and shoring up weak spots.
- Practice Final (O-practice-final-week-16, the paired practice exam in the Week 16 module) — a full, fresh, timed run that mirrors the real format and the 25-item emphasis. Best for pacing and a final readiness check.

(This guide points to both on purpose — it doesn't duplicate them.)

How the final is graded + test-taking strategy

How it's graded.
- 100 points across 25 items, weighted toward application (doing/interpreting, not reciting). Partial credit is available on multi-step numeric and "read-the-output" items where shown — show your work so you can earn it.
- The final is 30% of your course grade — the largest single assessment. It replaces Week 16's quiz, assignment, and discussion (there are none that week).
- Coverage matches this guide: Obj 1 ≈ 2 · Obj 2 ≈ 3 · Obj 3 ≈ 2 · Obj 4 ≈ 4 · Obj 5 ≈ 3 · Obj 6 ≈ 4 · Obj 7 ≈ 4 · Obj 8 ≈ 3. The inference block (Obj 6–8) is ~11 of 25 — practice it until the procedures are automatic.

Honest test-taking strategies for this material.
1. Name the shape before you pick a summary. Skew or an outlier → the answer wants the median/IQR, not the mean/SD.
2. Circle the keyword in probability items: and / or / given / at least / exactly / independent / mutually exclusive. "At least one" almost always means 1 − P(none); "given" picks the conditional denominator.
3. Run BINS before any binomial formula, and check the normal-approx license (np ≥ 10 and n(1−p) ≥ 10) before reaching for the bell.
4. For a sample mean, divide by σ/√n, not σ. The √n is the single most common slip on CLT and inference items.
5. Match the interval to the parameter: mean → t (df = n−1); proportion → z (no df). Don't use 1.96 for a small-sample mean.
6. Never write "reject H₀" alone. Finish with the sentence about the world; and remember significant = probably real, not big, and never causal.
7. For regression: observed − predicted for residuals, slope carries units, r² is a percent of variation, and never extrapolate beyond the data's x-range.
8. For every inference item, run the four beats: State → Compute → Compare → Conclude. Writing them as a checklist prevents skipped steps.
9. Sanity-check every probability and proportion: it must land in [0, 1]. A negative or >1 answer means a missed overlap or a mis-set denominator.
10. Do the easy items first, flag the hard ones, and budget your time — 25 items in the period is a few minutes each. Don't sink ten minutes into one item while four quick ones wait. Read each item twice and answer the question actually asked.

Canvas placement block

canvas_object   = Page
title           = "Final Exam Study Guide — Weeks 1–15 (Objectives 1–8)"
module          = "Week 16 — Final Review & Exam"
grading_type    = not_graded
available_from  = 2026-12-07      # posts before the Week 16 final exam window opens
published       = true
provenance      = "~ Prof. Rivera's edition · Fall 2026 · built with thecoursemaker.com"

Term-update note: each term's $39 update regenerates fresh practice variants from this same scope — the live final is never reproduced here.

The per-term $39 update (fresh assessment variants, re-paced to your next calendar) referenced above is on the roadmap — coming soon. Today's download is yours to keep, but it doesn't refresh itself.

~ Prof. Rivera's edition · Fall 2026 · built with thecoursemaker.com