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Week 16 · Final study guide

Final Exam — Study Guide

Introduction to Statistics Free sample edition

Coverage — what the final covers

The final is cumulative: it reaches back across all fifteen teaching weeks and all eight course objectives. It is given in Week 16, the finals week, which has no new material. The exam has 40 questions, and every question counts equally. Most are multiple choice; a few are true/false, a few ask you to select every statement that applies, and two ask you to type a short number. There is no time limit, and you get one attempt.

The questions are drawn from a bank built on the skills of Weeks 5–7 and 9–15. The Week 1–4 ideas and the Week 8 z-score still matter: they come back inside those later questions. A question about a confidence interval can turn on whether the data are a sample or a population, and a question about the normal model can turn on skew.

The course blueprint weights the eight objectives like this. Put the most review where the weight is largest.

The final's eight objectives, the weeks that teach each, and the share of the exam the course blueprint assigns it
Objective What you do Weeks Blueprint share
Objective 1 (distinguish populations from samples and parameters from statistics, classify variables by type and level of measurement, and identify appropriate sampling methods and study designs, including what each can and cannot conclude) Tell a sample from a population and a statistic from a parameter; judge sampling methods and study designs 1, revisited in 10 and 12 10%
Objective 2 (summarize and display a single quantitative or categorical variable with appropriate tables and graphs, and describe its distribution's shape, center, and spread with the right numerical summaries) Describe one variable's shape, center, and spread; place a value with a z-score 2, 3, 8 10%
Objective 3 (describe the relationship between two variables using scatterplots, the correlation coefficient, and two-way tables, and explain why association alone does not establish causation) Describe two variables with a scatterplot, r, and a two-way table; keep association apart from causation 4, revisited in 15 10%
Objective 4 (apply the basic rules of probability, including conditional probability and independence, and compute and interpret the expected value and standard deviation of a discrete random variable, including binomial settings) Apply the probability rules; find an expected value; work a binomial setting 5, 6, 7 10%
Objective 5 (use the normal model and sampling distributions, including the Central Limit Theorem, to reason about variability in individual values and in sample statistics) Use the normal model and sampling distributions, including the Central Limit Theorem 7, 9, 10 15%
Objective 6 (construct and correctly interpret confidence intervals for a population mean and a population proportion, including the meaning of the confidence level and the margin of error) Build and interpret confidence intervals for a mean and a proportion 11, 12 15%
Objective 7 (carry out and correctly interpret hypothesis tests for means and proportions, including one- and two-sample settings, p-values, significance levels, and Type I and Type II errors) Carry out and interpret hypothesis tests; name Type I and Type II errors 13, 14 20%
Objective 8 (fit and interpret a simple linear regression model, assess its fit with r-squared and residuals, and carry out inference for the slope) Fit and interpret a regression line; test and estimate the slope 15 10%

Inference, Weeks 11–14, carries more than a third of the blueprint on its own. Because the questions are drawn from a bank, the share on any one objective can differ from this plan. Inference and the normal model carry most of the exam either way.

Topic-by-topic review

Each unit below gives the idea in plain words first, then the course's exact definitions, then the mistake students make most often and its cure. The "deeper review" line tells you where to look in that week's materials.

Unit 1 — Where the data came from (Week 1)

  • Key idea. Every statistics question starts with a group you care about and the part of it you actually measured. A population is the entire group of individuals that a question is about. A sample is the part of a population that is actually observed or measured.
  • Parameter versus statistic. A parameter is a number that describes a whole population. A statistic is a number computed from a sample; it estimates the matching parameter. The population mean is μ and the sample mean is x̄.
  • Sampling. A simple random sample gives every possible group of n individuals the same chance. Stratified, cluster, and systematic samples are the other designed methods. A convenience sample and a voluntary-response sample build in bias: a built-in tendency to miss the truth in the same direction every time.
  • Study design. In an observational study, researchers record what they find without assigning any treatment. Only a randomized experiment supports a cause-and-effect conclusion.
  • Predictable mistake → cure. "A bigger sample fixes a biased method." It does not: size shrinks sampling variability, never bias. Cure: ask how the individuals were chosen before you ask how many.
  • Deeper review: the Week 1 lecture outline and chapter.

Unit 2 — Describing one variable (Weeks 2, 3, and 8)

  • Key idea. A distribution has a shape, a center, and a spread, and you report all three. A right-skewed distribution has its long tail toward larger values; a left-skewed one has its tail toward smaller values.
  • Center. The mean is the sum of all the values divided by how many values there are. The median is the middle value once the data are put in order. The median is a resistant measure; the mean is not. Skew pulls the mean toward the long tail.
  • Spread. The standard deviation is the square root of the variance, the typical distance of a value from the mean, in the data's own units. A sample divides the squared deviations by n − 1; a population divides by N.
  • Five-number summary and fences. The IQR is Q3 − Q1. The 1.5 × IQR rule flags a value beyond Q1 − 1.5 × IQR or Q3 + 1.5 × IQR as a potential outlier.
  • Relative standing (Week 8). A z-score is the number of standard deviations a value sits above or below the mean of its group: z = (x − x̄) ⁄ s.
  • Predictable mistakes → cures. Naming the skew by where the pile is: name it by where the tail is. Reporting the variance in the data's units: variance is in squared units, so take the square root. Averaging category codes: the mean needs interval or ratio data.
  • Deeper review: the Week 3 outline's month-length and penguin examples; the Week 8 outline's two 4,000 g penguins.

Unit 3 — Two variables (Week 4)

  • Key idea. A scatterplot shows direction, form, and strength. The correlation coefficient r is a unitless number between −1 and 1 that measures the direction and strength of a straight-line relationship.
  • Two-way tables. A conditional distribution keeps only one row or column and divides by that row's or column's total. Two categorical variables are associated when the conditional distributions differ.
  • Association is not causation. A lurking variable influences both variables and can create an association between them.
  • Predictable mistakes → cures. "r near 0 means no relationship": it means no linear relationship, so look at the plot. "Correlation proves cause": ask whether the data came from a randomized experiment.
  • Deeper review: the Week 4 outline (r = 0.8712 for flipper length and body mass).

Unit 4 — Chance (Weeks 5, 6, and 7)

  • Key idea. A probability is the long-run share of repetitions in which an event happens, a number from 0 to 1.
  • The rules. The complement rule: P(not A) = 1 − P(A). The general addition rule: add the two probabilities, then subtract the probability that both happen. Conditional probability: keep only the outcomes in the second event and find the share of them in the first. Two events are independent when learning that one happened leaves the probability of the other unchanged.
  • Random variables. The expected value is the probability-weighted average of a random variable's values; it is the long-run average and need not be a possible value. Shifting moves the mean and leaves the standard deviation alone; rescaling multiplies both.
  • The binomial. The binomial setting has a fixed number of trials n, independent trials, two outcomes, and the same chance of success p. Its mean is np and its standard deviation is √(np(1 − p)). The course's large-counts rule of thumb asks for np and n(1 − p) both at least 10 before a normal curve stands in.
  • Predictable mistakes → cures. Mixing up mutually exclusive and independent: mutually exclusive events can never both happen, so learning one happened changes the other's chance to 0. Reading P(A | B) as P(B | A): say which group you are keeping. The gambler's fallacy: a die has no memory.
  • Deeper review: the Week 5 penguin two-way table; the Week 7 guessing-quiz table.

Unit 5 — The normal model and sampling distributions (Weeks 9 and 10)

  • Key idea. A normal model is a bell-shaped distribution you assume describes a real variable, set by its mean μ and standard deviation σ. The empirical rule says about 68%, 95%, and 99.7% of values lie within one, two, and three standard deviations; it fits bell-shaped data, not skewed data.
  • Forward and inverse. A forward calculation standardizes a value with z = (x − μ) ⁄ σ and finds an area. An inverse calculation starts from an area and returns a value with x = μ + z · σ.
  • Sampling distributions. A statistic changes from sample to sample. The sampling distribution of x̄ has mean μ and standard error σ ⁄ √n. The Central Limit Theorem says the distribution of x̄ is approximately normal when n is large, whatever the population's shape. The sample proportion p̂ has mean p and standard deviation √(p(1 − p) ⁄ n).
  • Predictable mistakes → cures. "The CLT makes the data normal": it is the distribution of the mean that becomes approximately normal. Confusing σ with σ ⁄ √n: σ describes single values, and the standard error describes averages.
  • Deeper review: the Week 9 commute model; the Week 10 fair-die averages.

Unit 6 — Confidence intervals (Weeks 11 and 12)

  • Key idea. An interval estimate is a point estimate plus or minus a margin of error: the critical value times the standard error. The margin measures sampling variability only, not bias.
  • The two intervals. For a mean: x̄ ± t* · s ⁄ √n, with df = n − 1. For a proportion: p̂ ± z* · √(p̂(1 − p̂) ⁄ n), after checking for at least 10 successes and 10 failures.
  • The confidence level is the long-run share of samples for which the method captures the parameter. It describes the method, not the chance that one computed interval is right.
  • Planning a poll. Use n = (z* ⁄ ME)² × p̂(1 − p̂) with the conservative p̂ = 0.5, and always round up.
  • Predictable mistakes → cures. "There is a 95% probability μ is in this interval": the 95% belongs to the procedure. "95% of the individuals are in the interval": the interval describes μ, not single values.
  • Deeper review: the Week 11 Adelie interval, 3,626.92 g to 3,774.40 g; the Week 12 poll ladder.

Unit 7 — Hypothesis tests (Weeks 13 and 14) — the heaviest unit

  • Key idea. A test assumes the null hypothesis H₀ is true and asks how surprising the data would be. The alternative hypothesis Hₐ is the claim the test looks for evidence of.
  • The p-value is the probability, computed assuming the null hypothesis is true, of getting data at least as extreme as the data observed. It is not the probability that the null hypothesis is true.
  • Decisions. Reject H₀ when the p-value is at or below α; otherwise fail to reject, which is not the same as showing H₀ is true. A Type I error rejects a true null; its probability is α. A Type II error fails to reject a false null; its probability is β, and power is 1 − β.
  • The four procedures. One mean: the one-sample t test, t = (x̄ − μ₀) ⁄ (s ⁄ √n). Two independent means: the two-sample t test, with Welch's degrees of freedom read from software. One proportion: the one-proportion z test, with the null value p₀ inside the standard error. Two proportions: the two-proportion z test, with the pooled proportion.
  • Tests and intervals agree. A two-sided test at level α rejects a null value exactly when that value lies outside the matching 1 − α confidence interval.
  • Predictable mistakes → cures. "A smaller p-value means a bigger effect": size is a separate, practical question. "Not significant means no difference": it means the data are consistent with no difference.
  • Deeper review: the Week 13 guessing-quiz test; the Week 14 penguin tests.

Unit 8 — Regression (Week 15)

  • Key idea. The least-squares regression line ŷ = b₀ + b₁x is the line that makes the sum of the squared residuals as small as possible. The slope b₁ gives the predicted change in y for each one-unit increase in x.
  • Residuals and fit. A residual is y − ŷ. The coefficient of determination r² is the share of the variation in the response that the line accounts for; it is not the share of points on the line.
  • Inference for the slope. The t test for the slope computes t = b₁ ⁄ SE(b₁) on n − 2 degrees of freedom. The interval is b₁ ± t* × SE(b₁).
  • Predictable mistakes → cures. "The slope is the correlation": the slope carries units. Extrapolation: check the observed range of x before you predict. A significant slope in observational data shows association, not cause.
  • Deeper review: the Week 15 penguin line, ŷ = −5,780.8314 + 49.6856x.

Representative practice

These are fresh variants, not the exam's questions. Each worked example uses the course's own datasets with a new question, so the numbers stay pinned while the task changes. Work each one, then check it.

Worked example 1 — one weight, three species (Units 1 and 2)

Week 8 placed a 4,000 g Adelie at z ≈ 0.65 and a 4,000 g Gentoo at z ≈ −2.13. Where does a 4,000 g Chinstrap stand? The 68 Chinstrap penguins in the field study have mean 3,733.09 g and sample standard deviation 384.34 g.

  1. Distance from the Chinstrap mean: 4,000 − 3,733.09 = 266.91 g.
  2. Divide by the standard deviation: z = 266.91 ⁄ 384.34 ≈ 0.69, rounded to two decimals.

The bird sits about 0.69 standard deviations above the Chinstrap mean, an ordinary, slightly heavy Chinstrap. Why it matters: one weight tells three stories, because a z-score always depends on its reference group. The 68 birds are a sample, so 3,733.09 g is a statistic.

Worked example 2 — a sum of 8, and a longer guessing quiz (Unit 4)

Two dice. Five of the 36 equally likely ordered outcomes give a sum of 8: (2, 6), (3, 5), (4, 4), (5, 3), and (6, 2). So P(sum = 8) = 5 ⁄ 36 ≈ 0.1389, and by the complement rule P(sum ≠ 8) = 31 ⁄ 36 ≈ 0.8611, each rounded to four decimals.

A 40-question guessing quiz. A student guesses on 40 four-option questions, so n = 40 and p = 0.25.

  1. Mean: np = 40 × 0.25 = 10 correct answers.
  2. Standard deviation: √(40 × 0.25 × 0.75) = √7.5 ≈ 2.7386, about 2.74 correct answers.
  3. Rule of thumb: np = 10 and n(1 − p) = 40 × 0.75 = 30 are both at least 10, so a normal curve is a reasonable stand-in.

Over many guessed 40-question quizzes, the average is 10 right, and a typical score lands about 2.74 away from 10. Why it matters: the 10-question version failed the rule of thumb (np = 2.5), and quadrupling the questions is exactly what moves np up to 10.

Worked example 3 — one commute and nine commutes (Unit 5)

Suppose one-way commute times follow a normal model with μ = 32 minutes and σ = 6 minutes, the Week 9 model.

  1. Between 26 and 44 minutes. 26 is one standard deviation below the mean and 44 is two above. Half of the middle 68% lies between 26 and 32, which is 34%. Half of the middle 95% lies between 32 and 44, which is 47.5%. Together: 34% + 47.5% = 81.5%.
  2. The average of 9 trips. The standard error is σ ⁄ √n = 6 ⁄ √9 = 6 ⁄ 3 = 2 minutes.
  3. A 9-trip average above 36 minutes. z = (36 − 32) ⁄ 2 = 2, and by the empirical rule about 2.5% of 9-trip averages land above 36 minutes.

About 81.5% of single trips take between 26 and 44 minutes, and only about 2.5% of nine-trip averages run longer than 36 minutes. Why it matters: averages wobble less than single values, by the factor √n.

Worked example 4 — two intervals (Unit 6)

A mean, with a smaller sample. What if only 25 Chinstrap penguins had been weighed, with the same mean of 3,733.09 g and standard deviation of 384.34 g?

  1. Degrees of freedom: df = 25 − 1 = 24, so t* for 95% is 2.064.
  2. Standard error: SE = 384.34 ⁄ √25 = 384.34 ⁄ 5 = 76.868 g.
  3. Margin of error: 2.064 × 76.868 ≈ 158.66 g.
  4. Interval: 3,733.09 − 158.66 = 3,574.43 g and 3,733.09 + 158.66 = 3,891.75 g.

We are 95% confident that the population mean body mass of Chinstrap penguins like these is between 3,574.43 g and 3,891.75 g.

A proportion, at 90%. The female share among the 333 sexed penguins is p̂ = 0.4955 with SE = 0.0274. At 90%, z* = 1.645, so the margin is 1.645 × 0.0274 ≈ 0.0451, and the interval runs from 0.4955 − 0.0451 = 0.4504 to 0.4955 + 0.0451 = 0.5406.

Why it matters: the 90% interval, 0.4504 to 0.5406, is narrower than the 95% interval, 0.4418 to 0.5492. Less confidence buys a narrower range.

Worked example 5 — two tests and the intervals that match them (Unit 7)

A mean. Is the Adelie mean body mass equal to a claimed 3,800 g? H₀: μ = 3,800 g; Hₐ: μ ≠ 3,800 g. With x̄ = 3,700.66 g and SE = 37.32 g on df = 150:

  1. t = (3,700.66 − 3,800) ⁄ 37.32 = −99.34 ⁄ 37.32 ≈ −2.66.
  2. The size of t, 2.66, is beyond t* = 1.976, so reject H₀ at α = 0.05.
  3. Check with the interval: 3,800 g lies outside 3,626.92 g to 3,774.40 g, which matches the rejection.

A proportion. Is the female share different from 0.45? H₀: p = 0.45; Hₐ: p ≠ 0.45. The null standard error uses p₀: √(0.45 × 0.55 ⁄ 333) ≈ 0.0273.

  1. z = (0.4955 − 0.45) ⁄ 0.0273 = 0.0455 ⁄ 0.0273 ≈ 1.67.
  2. At α = 0.05, 1.67 is inside ±1.960, so fail to reject. At α = 0.10, 1.67 is beyond 1.645, so reject.
  3. Check with the intervals: 0.45 sits inside the 95% interval (0.4418 to 0.5492) and just outside the 90% interval (0.4504 to 0.5406).

Why it matters: the same data can reject at one α and not at another, which is why α is chosen before the data are seen.

Worked example 6 — a prediction, a residual, and a warning (Unit 8)

The penguin line is ŷ = −5,780.8314 + 49.6856x, with x the flipper length in mm and observed flippers from 172 mm to 231 mm.

  1. Predict at 180 mm. 49.6856 × 180 = 8,943.408, and 8,943.408 − 5,780.8314 ≈ 3,162.58 g. Since 180 mm is inside 172 to 231 mm, this is interpolation.
  2. A residual. Suppose a hypothetical bird with a 180 mm flipper weighs 3,400 g. Its residual is 3,400 − 3,162.58 = 237.42 g, so it is about 237 g heavier than predicted.
  3. Twenty more millimetres. 20 × 49.6856 ≈ 993.71 g more predicted body mass, about one kilogram.
  4. A warning. At 160 mm, the line gives 49.6856 × 160 − 5,780.8314 ≈ 2,168.86 g. But 160 mm is below the smallest observed flipper of 172 mm, so this is extrapolation and the line gives no warrant for it.

Why it matters: a line describes the data it was built from; the range of x is the first thing to check.

Self-check questions

  1. A music-streaming app posts a pop-up poll asking whether its new layout is better, and 12,000 listeners tap a reply. Does the large count make the result trustworthy? Self-check: No. This is a voluntary-response sample, so it builds in bias, and a bigger sample does not remove bias. Why: the listeners with the strongest feelings choose to reply.
  2. In a leap year February has 29 days, so the twelve month lengths add to 366. What happens to the mean and the median? Self-check: the mean rises to 366 ⁄ 12 = 30.5 days; the median stays 31. Why: the median depends only on the middle values, and February is not one of them.
  3. The flipper-mass correlation is r = 0.8712 with mass in grams. What is r with mass in kilograms? Self-check: still 0.8712. Why: r carries no units, so changing units leaves it unchanged.
  4. From the species-by-island table, find P(Adelie | Dream) and compare it with P(Adelie). Self-check: P(Adelie | Dream) = 56 ⁄ 124 ≈ 0.4516 and P(Adelie) = 152 ⁄ 344 ≈ 0.4419. Why: the two differ, so learning the island changes the chance, though here only a little.
  5. A game pays two tokens per pip on one fair die, so the payout is 2X. Find its mean and standard deviation. Self-check: the mean is 2 × 3.5 = 7 tokens and the standard deviation is 2 × 1.7078 = 3.4156 tokens. Why: rescaling multiplies both the mean and the standard deviation.
  6. On the 10-question guessing quiz, find P(X ≤ 3). Self-check: P(X ≤ 2) + P(X = 3) = 0.5256 + 0.2503 = 0.7759. Why: "at most 3" adds every count from 0 up to 3.
  7. A random sample of 625 people is drawn from a population where p = 0.5. How much does p̂ typically wobble? Self-check: √(0.5 × 0.5 ⁄ 625) = 0.02, so about 95% of such samples land within 2 × 0.02 = 0.04 of 0.5. Why: the spread of p̂ shrinks with the square root of n.
  8. How many people does a poll need for a 95% margin of error of ±2 percentage points? Self-check: 1.96² × 0.25 ⁄ 0.02² = 0.9604 ⁄ 0.0004 = 2,401 people. Why: tightening the target from 3 points (1,068 people) to 2 points more than doubles the sample, because the margin shrinks only with the square root of n.
  9. An airport scanner tests H₀: "this bag holds no banned item." What are the two errors? Self-check: a Type I error flags a bag that is actually clean; a Type II error lets a bag with a banned item through. Why: Type I rejects a true null, and Type II fails to reject a false one.
  10. A promoter compares the share of tickets sold in the first hour at two different concerts. Which procedure fits? Self-check: a two-proportion z test, using the pooled proportion. Why: the question compares two proportions from two independent groups.

Check yourself — choose one. (a) Which unit in this guide felt least familiar, and which one worked example would you redo first? (b) Pick two units that connect, such as the normal model and confidence intervals, and explain the link in two sentences. Either choice is a good use of ten minutes.

Watch before the exam

These six short videos come from the course's own weekly readings. Use them for review and for connecting weeks, not for new material.

Study plan

A short countdown sized to a course that meets twice a week. Count backward from the day the final opens in Week 16.

  • Two weeks out: re-read the Week 1–4 outlines and redo Worked examples 1 and 2 from scratch. These ideas return inside later questions.
  • Ten days out: review Units 4 and 5. Redo Worked example 3 until the empirical-rule steps feel automatic.
  • One week out: review Units 6 and 7, the heaviest part of the blueprint. Redo Worked examples 4 and 5 without looking.
  • Five days out: take the paired practice exam once, closed-book and under exam-like conditions, then sort every miss by unit.
  • Three days out: run the paired exam-prep tutorial (the copy/paste chatbot tutor, or its no-AI route) on the two units your practice exam showed were weakest.
  • Two days out: review Unit 8 and Worked example 6, then retake the practice exam.
  • The day before: skim the "Predictable mistake → cure" lines in every unit and watch one video from the list above. Sleep.

Use the exam-prep tutorial and the practice exam for active recall; this guide is for review, and those two are where you rehearse. Rereading alone moves scores less than retrieval practice does.

Exam format and scoring

The final is scored automatically, and all 40 questions count equally. The questions appear one at a time, and you cannot go back to an earlier question once you move on. You have one attempt and no time limit. AI tools are not permitted on the final.

Test-taking strategies for this material:

  • Before you compute, name the tool: one mean, two means, one proportion, two proportions, or a slope. The procedure follows from the question and the data type.
  • Say every interval in its sentence: "we are 95% confident that the population mean is between ___ and ___."
  • Say every p-value in its sentence: "if the null hypothesis were true, data this extreme would occur about ___ of the time."
  • Check sample versus population before you choose a symbol or a divisor.
  • Check the observed range of x before you trust a regression prediction.
  • Since you cannot go back, settle each question before you move on.

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