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Week 16 · Final study guide

Final Exam — Study Guide

Critical Thinking Free sample edition

Coverage — what the final covers

The final is cumulative over Weeks 1–15. It opens in Week 16, the finals week, and closes at the due time your instructor sets. There is no time limit, and you get one attempt. The questions appear one at a time, and you cannot go back once you move on.

Format: 40 questions, and every question counts the same. Thirty-eight ask you to pick the one best option, and two ask you to judge a statement as true or false. Every passage, table, and chart a question uses is written out in the question in words. A few questions ask for a small calculation, such as a percent from a table, and scratch paper is enough for it.

What is in scope, with the number of final questions on each course objective:

Final coverage by course objective: the weeks that teach it, its main skills, and its number of questions out of 40
Course objective Weeks that teach it Main skills Questions
Identifying arguments (Objective 1) 1, carried into 11 Claims; argument vs. explanation; premises and conclusions with and without indicator words 1
Analyzing arguments (Objective 2) 2, carried into 14 and 15 Standard form; the supplied premise; argument maps with objections and replies 3
Evaluating deductive reasoning (Objective 3) 3 to 5, carried into 9 and 13 Valid, sound, strong, cogent; the counterexample method; the conditional forms 3
Language and fallacies (Objective 4) 6 to 8, carried into 10 and 12 Definitions and wording; the fallacies of relevance and presumption; fallacy vs. false premise 4
Inductive reasoning (Objective 5) 9 and 10 Samples; analogies; correlation and causation; confounding factors; Mill's methods; post hoc and the slippery slope 8
Numbers in public claims (Objective 6) 11 Percentage points and percent; relative and absolute risk; base rates in natural frequencies; misleading graphs 3
Sources and evidence (Objective 7) 12 The four credibility questions; primary and secondary sources; tracing a claim; context; AI-made misinformation 5
Science and your own thinking (Objective 8) 13 and 14 Testability; controls; the best explanation; the appeal to ignorance; the biases and the debiasing habits 8
Values and building an argument (Objective 9) 15 Value and factual premises; the three frameworks; steelmanning; the course synthesis 5

Where the weight sits. All 40 questions come from the material of Weeks 9–15. Weeks 10, 12, and 15 carry the most, 7 questions each, so put the most review time there. Week 14 comes next with 6, then Week 13 with 5, and Weeks 9 and 11 with 4 each. By objective, the inductive tests (Objective 5) and science and bias (Objective 8) lead with 8 questions each. By difficulty, 16 questions are easier, 20 are medium, and 4 are harder.

The first half of the course still matters. Eleven questions check Objectives 1 to 4, and every one of them sits inside a later week's material. A hypothesis test uses Week 4's forms, a sampling question uses Week 3's strength and cogency, Week 7's fallacies return in Weeks 10 and 12, and Week 2's argument map returns in Weeks 14 and 15. So the early weeks get a short callback section below, and Weeks 9 to 15 get the full review.

Topic-by-topic review

Each topic gives the one distinction it asks for, the key idea in plain words, the course's own definitions, a course example, and the mistakes students make most often with their cures. Every passage and scenario below is illustrative, written for the course and attributed to no one, unless it is marked real. The deeper review for every topic is that week's chapter, lecture page, and slide deck.

Weeks 1–8 in brief — the tools the later weeks run on

These weeks carry no question of their own on the final, but their tools run inside every later question. Review each line until you could say it to a friend.

  • Week 1, claims and arguments: a claim is a sentence that is either true or false. An argument is a set of claims in which one or more, the premises, are offered as reasons to accept another, the conclusion. An explanation gives reasons why something already accepted is so. "Because" is a clue, not proof: it also introduces explanations, and many arguments carry no indicator word at all.
  • Week 2, standard form and the map: standard form puts each premise on its own numbered line (P1, P2, P3), then a line reading "So," and then the conclusion on its own line marked C. A supplied premise is an unstated premise the analyst writes out and marks "(supplied)". The principle of charity chooses the most reasonable reading the words allow and supplies the weakest premise that connects the stated premises to the conclusion. On an argument map, an objection is an arrow marked "against" pointing at the claim it attacks, and a reply is an arrow marked "against" pointing at the objection.
  • Week 3, the four standards: an argument is valid when it is impossible for all its premises to be true and its conclusion false, and sound when it is valid and every premise is true. An inductive argument is strong when its premises, if true, would make its conclusion likely, and cogent when it is strong and all its premises are true. Even a cogent argument can have a false conclusion. A counterexample is a case with the same form in which every premise is true and the conclusion is false.
  • Week 4, the conditional: "if P, then Q" is false in exactly one situation, P true and Q false. The valid forms are modus ponens, modus tollens, hypothetical syllogism, and disjunctive syllogism. Affirming the consequent (if P, then Q; Q; so P) and denying the antecedent (if P, then Q; not P; so not Q) are invalid. Only the contrapositive (if not Q, then not P) is equivalent to the original. In "if P, then Q", P is sufficient for Q and Q is necessary for P.
  • Week 5, Venn diagrams: a shaded region is empty, and an X sits in a region that holds at least one thing. A syllogism is valid when the premises alone already shade or mark the conclusion's region.
  • Week 6, words: a vague term has borderline cases; an ambiguous term has two distinct meanings that both fit. Equivocation lets one word shift meaning between steps. Loaded language carries a conclusion without arguing for it, while the facts it describes can stay true.
  • Week 7, relevance: a fallacy is a flaw in the reasoning; a false premise is a flaw in the content. The relevance test asks: if this premise were true, would it bear on whether the conclusion is true? A documented conflict of interest is a reason to weigh testimony with care, not a refutation.
  • Week 8, presumption: the false dilemma offers two options as the only ones when others exist; begging the question assumes in a premise what the conclusion is meant to show; the loaded question builds a disputed claim into its wording.
  • Predictable mistakes → cures: (1) valid read as true. Cure: validity is about form; ask whether the premises are true as a separate question. (2) the converse read as equivalent. Cure: only the contrapositive swaps and denies both parts. (3) "because" read as always marking a premise. Cure: ask whether the claim is still in doubt for the audience. (4) any criticism of a person called an attack on the person. Cure: run the relevance test. (5) a false premise called a fallacy. Cure: a fallacy is a flaw in how the premises connect.

Week 9 — Inductive generalization and analogy

  • The distinction: a sample vs. a population, and size vs. representativeness.
  • Key idea: a claim about a whole group rests on the part of the group someone actually looked at, so the first question is how that part was chosen.
  • Definitions: the population is the whole group a claim is about. The sample is the part of a population that is actually observed or asked. A representative sample resembles the population in the ways that matter to the question. A biased sample is chosen in a way that favors some members of the population over others. Nonresponse is the problem that arises when the people who choose to reply differ from the people who do not. A random sample is drawn so that every member of the population has the same chance of being picked, which is different from asking whoever happens to be nearby.
  • Fallacies and analogies: the hasty generalization draws a conclusion about a whole group from too few cases, or from cases that do not represent it. An argument from analogy reasons that two things alike in several known ways are also alike in a further way. It is stronger when the similarities bear on the conclusion and no relevant difference, a disanalogy, outweighs them.
  • Course example (illustrative): on a campus of 12,000 students, the doorway sample asked 400 students at the library entrance at 11 p.m., and 300 said yes, which is 300 ÷ 400 × 100 = 75 percent. The random sample drew 400 from the enrollment list, and 240 said yes, which is 60 percent. Both samples hold 400 students, so size cannot explain the gap; the method of choosing does.
  • Real case, taught in words: in 1936 the Literary Digest sent mock ballots to roughly 10 million people, mostly taken from phone books and lists of car owners, got roughly 2.4 million back, and forecast that Landon would win. Roosevelt won 523 electoral votes to 8, the figure the National Archives reports. Accounts differ on how much of the miss came from the lists and how much from nonresponse, so the course teaches both and keys neither.
  • Predictable mistakes → cures: (1) a large sample read as automatically representative. Cure: ask how it was chosen. (2) one vivid story read as a trend. Cure: count the cases and ask whether they are typical. (3) an analogy read as proof. Cure: list the similarities, then hunt for the disanalogy. (4) a random sample read as "random people I happened to ask". Cure: every member of the population needs the same chance. (5) a generalization called valid. Cure: it is inductive, so it is strong or weak, and even a cogent one can miss.

Week 10 — Causal reasoning

  • The distinction: correlation vs. causation, and a supported slope vs. an unsupported one.
  • Key idea: two things that go together may share a cause, may run the other way, or may line up by chance, so a pattern is the start of a causal check, not the end of it.
  • Definitions: a correlation is a pattern in which two things tend to go together; by itself it does not show that either one causes the other. A causal claim says one thing makes another happen. A confounding factor is a third factor that drives both of two things that go together. Reverse causation is the alternative in which the causal arrow runs the other way. Coincidence is a pattern that arises by chance, most likely when it rests on only a few cases.
  • Fallacies: after this, therefore because of this (post hoc) concludes that one event caused another only because it came first. A slippery slope predicts that one step will lead through a chain of further steps to a bad outcome. It is reasonable when each link is backed by evidence and a fallacy, the unsupported slippery slope, when the links are not.
  • Methods: Mill (1843), A System of Logic, sets out the methods of agreement, difference, residues, concomitant variation, and a joint method. The course works two. The method of agreement looks for the one factor that every case of an effect shares. The method of difference compares a case with the effect and a case without it that are alike in every factor but one. A controlled comparison holds other factors fixed, or spreads them evenly by random assignment, so a difference in outcome can be traced to the one factor that differs.
  • Course example (illustrative): in the hot-weather table, 28 of 50 high-ice-cream days had many sunburn cases, 56 percent, against 16 of 50 low-sales days, 32 percent. Within hot days the shares are 80 percent and 80 percent, and within mild days 20 percent and 20 percent. The gap disappears once temperature is held fixed, so hot weather is the third factor driving both.
  • Predictable mistakes → cures: (1) correlation read as causation. Cure: run the three alternatives before any causal verdict. (2) a confounding factor read as "an error in the data". Cure: a table can be counted perfectly and still be read wrongly. (3) every slippery slope called a fallacy. Cure: check whether each link has evidence. (4) after-this read as because-of-this. Cure: ask for a comparison that rules out the other explanations.

Week 11 — Numbers in the news

  • The distinction: percentage points vs. percent change, and relative vs. absolute risk.
  • Key idea: a true number can still mislead when it is reported without its partner, so the course always asks for both.
  • Definitions: percentage points measure the plain difference between two percents, found by subtraction. Percent change is the difference between a new value and a starting value, divided by the starting value and multiplied by 100. Absolute risk states the chance of an outcome as a count out of a named base, such as 2 in 1,000 people. Relative risk compares one risk with another as a ratio or a percent change; without the base it leaves out how many people are affected. A base rate is how common a condition is in the whole group before any test. Natural frequencies set out a probability problem as counts of people in a group of round size. Base-rate neglect judges what a positive result means from a test's accuracy alone.
  • Graphs: a truncated axis starts above zero, so bar heights stop matching the values. A stretched scale makes a small change look steep. A cherry-picked time window shows only the period that fits the speaker's story.
  • Course examples (illustrative): random-sample support rose from 50 percent to 60 percent: 60 − 50 = 10 percentage points, and (60 − 50) ÷ 50 × 100 = 20 percent. A risk rising from 2 in 1,000 to 3 in 1,000 is a 50 percent relative increase and 1 more case per 1,000. In the screening table, the condition affects 1 in 100 of the 1,000 people screened; 9 of the 10 with it test positive, and 99 of the 990 without it test positive anyway, so 9 of the 108 people who test positive have the condition. The ad's bars of 50 and 55 on an axis from 45 are drawn 5 and 10 units tall, yet the real gap is 10 percent.
  • Real case, taught in words: the National Cancer Institute explains that a 20 percent relative reduction in the risk of dying of lung cancer in a large screening trial amounted to an absolute reduction of about 0.4 percent. No figure from it is keyed.
  • Predictable mistakes → cures: (1) a percentage-point change written as a percent. Cure: subtract for points, divide by the start for percent, and report both. (2) a 50 percent relative rise read as large. Cure: ask "out of how many?" (3) a positive screening read as near-certain. Cure: build the natural-frequency table. (4) bar heights read as the size of the gap. Cure: check where the axis starts.

Week 12 — Evaluating sources and evidence

  • The distinction: a primary vs. a secondary source, and relevant expertise vs. authority outside the field.
  • Key idea: before you weigh what a source says, find out who is speaking, what they stand to gain, and where the claim started.
  • Definitions: credibility is how far a source deserves trust on a particular claim, judged by its relevant expertise in that field, its track record of accuracy, its independence from the outcome, and whether it shows the evidence behind the claim. A primary source is the original record of an event, study, or statement; a secondary source reports, summarizes, or interprets one. Tracing a claim follows it back through the sources that repeat it to the primary source, then checks that the original says what the retellings say. An independent source has no tie to the one being checked, and you consult it by leaving the page before reading closely.
  • Context and AI: a real photo, video, or quotation is out of context when it is shown apart from its original date, place, event, or surrounding words. A fabricated citation points to a work that does not exist, however correct its format. AI-generated misinformation includes a chatbot's invented statistic, study, quotation, or citation, and a cloned voice. The Federal Trade Commission's 2023 consumer alert on cloned voices (real) advised checking such a call through a contact route you trusted before the call came.
  • Authority, called back from Week 7: an appeal to authority is good inductive evidence when the authority has relevant expertise, experts broadly agree, and no serious conflict of interest exists. It becomes the fallacy when one of those is missing, most often when someone speaks outside their field.
  • Course example (illustrative): P4's cost traced back to its inputs is 2 staff × 8 hours × 5 nights × 20 dollars an hour = 1,600 dollars, so the stated total checks out against its inputs. The sleep claim in the running question passes through 2 secondary sources before it reaches 1 primary study.
  • Predictable mistakes → cures: (1) a professional-looking site read as credible. Cure: ask the four questions about this one claim. (2) expertise in one field read as authority in another. Cure: name the field of the claim. (3) a quotation read as accurate because it is in quotation marks. Cure: find the surrounding words. (4) a chatbot's citation read as real because it is formatted well. Cure: look it up in a catalog. (5) a source that flunks the check read as proof the claim is false. Cure: it shows only that this source has not earned your trust on it.

Week 13 — Science and explanation

  • The distinction: a failed prediction vs. a passed one, and "unproven" vs. "proven false".
  • Key idea: a scientific claim sticks its neck out, so you test it by looking for the observation that could show it wrong.
  • Definitions: a hypothesis is a proposed explanation still being tested, and a prediction is what you should observe if it is true, stated before you look. A testable claim is one that some possible observation could show to be wrong. Falsifiability is the property Popper proposed as the test that separates science from non-science; it is one influential proposal, and it has standard critics. A control group does not get the treatment; a placebo is a look-alike treatment with no active ingredient; blinding keeps people from knowing who got which. A scientific theory is a well-tested, broad explanation of many observations, not a guess.
  • The best explanation: inference to the best explanation reasons to the hypothesis that would best explain the evidence, judged by fit, simplicity, scope, and consistency with what else is known. Harman (1965), "The Inference to the Best Explanation", The Philosophical Review, gave the phrase; Peirce's older name for the move is abduction.
  • Testing and its logic: if a prediction fails, the reasoning "if the hypothesis is true, the prediction comes true; the prediction did not come true; so the hypothesis is false" is modus tollens, a valid form. If the prediction comes true, concluding that the hypothesis is true affirms the consequent, so a passed test counts in the hypothesis's favor but cannot settle it.
  • Pseudoscience and proof: the five warning signs of pseudoscience are no possible result could count against the claim, testimonials stand in for controlled tests, methods and data are kept secret, persecution is claimed in place of evidence, and the claim never changes despite contrary evidence. They are signs to investigate, not proofs. The appeal to ignorance (ad ignorantiam) concludes a claim is true because no one has shown it false, or false because no one has shown it true. The burden of proof lies with the person who makes the claim.
  • Course example (illustrative): in the course's supplement trial, 51 of 60 in the supplement group and 51 of 60 in the placebo group felt better, 85 percent each, so the supplement showed no edge over the look-alike pill, and the ad's 12 testimonials showed only that colds end.
  • Predictable mistakes → cures: (1) "theory" read as a mere guess. Cure: in science a theory is well tested and broad. (2) "unfalsifiable" read as "proven". Cure: a claim no observation could touch is untested. (3) one success story read as a test. Cure: ask where the comparison group is. (4) a lack of disproof read as proof. Cure: the burden stays with the claimer. (5) a passed prediction read as proof. Cure: concluding the hypothesis from it affirms the consequent.

Week 14 — Bias and decision-making

  • The distinction: a confirming test vs. a disconfirming test, and a heuristic vs. an error.
  • Key idea: the tests you ran outward all term now point inward, because your own reasoning tilts too.
  • Definitions: confirmation bias is the habit of seeking and favoring evidence that fits what one already believes. Wason (1960) showed participants the triple 2, 4, 6 as one case of a hidden rule and had them propose triples to find it; the rule turned out to be any rising sequence, and most people tested triples that fit their own guess. A confirming test should come out in a guess's favor, so a pass adds little; a disconfirming test could show the guess wrong. Motivated reasoning judges evidence more leniently when it supports a wanted conclusion. Overconfidence holds a belief or prediction with more certainty than the evidence supports.
  • Heuristics: a heuristic is a mental shortcut that often works and sometimes misleads. Tversky and Kahneman (1974), "Judgment under Uncertainty: Heuristics and Biases", Science, described the availability heuristic (judging how common something is by how easily examples come to mind) and anchoring (an estimate that stays too close to a starting number, the anchor).
  • Debiasing habits: consider the opposite, look for the case that would disprove your view, write the prediction down before seeing the result, and decide in advance what evidence would change your mind. The same-standard test asks whether you would accept a piece of evidence if it pointed the other way.
  • Course examples (illustrative): a jacket tagged "was 200 dollars, now 120" is 200 − 120 = 80 dollars off, and 80 ÷ 200 × 100 = 40 percent; whether 120 dollars is a fair price is a separate question. A team sure of a 6-day project had taken 9, 10, and 8 days before, an average of 27 ÷ 3 = 9 days, 3 days over the plan.
  • Predictable mistakes → cures: (1) confirmation bias read as something only other people have. Cure: audit your own evidence list. (2) a heuristic read as always wrong. Cure: it is a shortcut that often works. (3) knowing about a bias read as protection. Cure: use a habit, not a feeling. (4) more confirming tests read as stronger proof. Cure: run one test that could fail.

Week 15 — Reasoning about values and the course synthesis

  • The distinction: a value premise vs. a factual premise, and a steelman vs. a straw man.
  • Key idea: a moral argument pairs a standard for what ought to be with a description of what is, and each half needs its own kind of support.
  • Definitions: a moral argument concludes what someone ought to do or what is good, right, or required. A value premise says what is good, right, or required and is supported by reasons and principles. A factual premise says what the situation is and is supported by evidence. A value disagreement can remain even when both sides agree on every fact; a factual disagreement is about what the situation is, and evidence could settle it. Many moral disagreements turn out to be factual.
  • The three frameworks, as tools: the consequence-based framework judges an action by its outcomes for everyone affected; critics note that outcomes are hard to predict and that it can seem to allow harming a few for a gain to many. The duty-based framework judges an action by whether it keeps duties or rules, some holding regardless of outcomes; critics note that duties can conflict and that keeping a rule can lead to a bad outcome. The character-based framework asks what a person of good character would do; critics note that it can give little specific guidance in a hard case. The course gives no verdict on which framework is correct.
  • Steelmanning and the synthesis: steelmanning states the opposing view in its strongest form, ideally in terms its holders would accept, before replying; it is the opposite of a straw man and does not mean agreeing. The course synthesis matches each kind of argument with its test: the counterexample method for a claim that must follow, a Venn diagram for an "all" or "some" claim, a precise definition for a claim that hangs on a word, the fallacy check, the sampling check, the confounding check, percentage points and natural frequencies, the source check, testability and the best-explanation test, the bias check, and the value-and-fact split.
  • Course example (illustrative): in the running question, P3 (supplied) says what the campus should do, so it is the 1 value premise, and P1, P2, and P4 are the 3 factual premises. A reader can accept both P3 and the objection's value premise at once; the sides part ways on whether an open library adds all-night studying or only moves it, a factual question.
  • Predictable mistakes → cures: (1) a moral claim read as mere opinion beyond reasons. Cure: split it into its value and factual premises and weigh each. (2) one framework read as the correct one. Cure: each is a tool with a strongest case and standard objections. (3) steelmanning read as agreeing. Cure: you state the view at its best so your reply answers the real thing. (4) a factual disagreement read as a value one. Cure: ask whether evidence could settle it.

Representative practice

These are fresh variants, not the final's questions. Work each worked example yourself first, then compare your steps. Every scenario below is illustrative, invented for this guide, and attributed to no one. Every number is computed and checked in advance.

Worked example — a weekend bus survey and two analogies (Week 9)

A regional bus agency (illustrative) wants to know how many of its 16,000 rider-card holders want weekend service.

  1. Sample A: 320 riders asked aboard the late-evening route; 256 say yes. That is 256 ÷ 320 × 100 = 80 percent.
  2. Sample B: 320 card holders drawn at random from the full card list; 144 say yes. That is 144 ÷ 320 × 100 = 45 percent.
  3. Size: each sample is 320 of 16,000, and 16,000 ÷ 320 = 50, so each is 1 rider in 50. Size cannot explain the gap.
  4. Method: Sample A asked only people already riding late, the riders likeliest to want more service, so it is a biased sample. Sample B gave every card holder the same chance.
  • The verdict: Sample B's 45 percent is the better guide to the 16,000 riders. The 80 − 45 = 35 percentage-point gap comes from how Sample A was chosen.
  • Two analogies: "The new model of this blender has the same motor and the same maker as the last one, so its motor will probably last about as long." The similarities bear on how long a motor lasts, so the analogy is fairly strong. "A choir is like a sports team, so the director should cut whoever sings the fewest solos." Both are groups that practice, but that similarity does not bear on who belongs in a choir, so it is a weak analogy.

Why it matters: a generalization is only as good as the way its sample was chosen, and an analogy is only as good as the similarities that bear on its conclusion.

Worked example — hot cocoa and slips on the sidewalk (Week 10)

A winter town (illustrative) logs 60 days by hot-cocoa sales at a downtown stand (high or low) and by slip-and-fall reports on the main street (many or few):

An illustrative 60-day winter table: cocoa-stand sales against slip reports, split by icy and dry days
Days High sales, many slips High sales, few Low sales, many slips Low sales, few
Icy (20) 12 3 4 1
Dry (40) 1 9 3 27
All (60) 13 12 7 28
  1. Overall: high-sales days are 13 + 12 = 25, and 13 ÷ 25 × 100 = 52 percent had many slips. Low-sales days are 7 + 28 = 35, and 7 ÷ 35 × 100 = 20 percent did. That is a correlation.
  2. Icy days held fixed: 12 ÷ (12 + 3) × 100 = 12 ÷ 15 × 100 = 80 percent on high-sales days, and 4 ÷ (4 + 1) × 100 = 4 ÷ 5 × 100 = 80 percent on low-sales days.
  3. Dry days held fixed: 1 ÷ (1 + 9) × 100 = 1 ÷ 10 × 100 = 10 percent, and 3 ÷ (3 + 27) × 100 = 3 ÷ 30 × 100 = 10 percent.
  4. Why the gap appeared: 15 of the 25 high-sales days were icy, and 15 ÷ 25 × 100 = 60 percent; 5 of the 35 low-sales days were icy, and 5 ÷ 35 is about 14 percent, well under a fifth.
  • The verdict: the gap disappears once the weather is held fixed, so icy weather drives both cocoa sales and slips. Cocoa does not make people fall.
  • Three more checks: "Cafés that play louder music have more customers" may run backward, since a crowded café turns the music up to be heard over the noise (reverse causation). "The bakery switched ovens in the spring and its bread won a ribbon in the summer, so the oven won the ribbon" rests on order alone (post hoc). "If the club lets each member bring one guest, soon they will bring ten, then strangers will run the club" offers no evidence for any link (an unsupported slippery slope).
  • Mill's methods: three office printers (illustrative) jam on the same morning; they differ in age, model, and floor, and they share only a new box of paper. That is the method of agreement, and it points at the paper. Two identical seedlings on one sill get the same soil and water, and only the one moved under a grow light sprouts a second leaf by week's end. That is the method of difference.

Why it matters: a two-way table can show a real pattern and still mislead until you hold the third factor fixed.

Worked example — a market poll, a sprain risk, a screening, and a graph (Week 11)

  1. Points and percent (illustrative): support for a weekend farmers' market rose from 30 percent to 45 percent of residents polled. That is 45 − 30 = 15 percentage points, and (45 − 30) ÷ 30 × 100 = 50 percent. A post that says "support jumped 50 percent" is true but tells half the story.
  2. Relative and absolute (illustrative): a recreation league's yearly risk of an ankle sprain falls from 5 in 1,000 players to 4 in 1,000 after a new warm-up. The absolute change is 5 − 4 = 1 fewer sprain per 1,000 players, and the relative change is (5 − 4) ÷ 5 × 100 = 20 percent.
  3. Base rates in natural frequencies (illustrative): a test for a condition that 1 in 200 people have is given to 4,000 people.
An illustrative screening of 4,000 people in natural frequencies, with every count computed from the base rate
Group Test positive Test negative Total
Have the condition 18 2 20
Do not have it 199 3,781 3,980
Total 217 3,783 4,000
  • 4,000 ÷ 200 = 20 people have the condition; the test catches 18 and misses 20 − 18 = 2.
  • 4,000 − 20 = 3,980 do not; the test wrongly flags 5 percent of them, 3,980 × 5 ÷ 100 = 199, and 3,980 − 199 = 3,781 test negative.
  • 18 + 199 = 217 test positive, so 18 of the 217 people who test positive have the condition. In words: fewer than 1 in 12 of the positives is real, because the condition is rare.
  1. The truncated graph (illustrative): an ad compares two delivery apps rated 70 and 77 on an axis that starts at 63. The bars are drawn 70 − 63 = 7 and 77 − 63 = 14 units tall, so one looks twice the other. The real gap is 77 − 70 = 7, and 7 ÷ 70 × 100 = 10 percent.

Why it matters: every one of these numbers is true, and every one misleads until you report its partner: points with percent, relative with absolute, and positives against the base rate.

Worked example — a running-shoe claim and a help-room budget (Week 12)

  1. The trace (illustrative): a social post says a running shoe "cuts knee injuries in half" and credits a fitness blog. The blog credits a news story, and the news story describes a study. The chain holds 3 links: 2 secondary sources and 1 primary source, the study itself. Reading the study shows whether it measured knee injuries at all, in which runners, and how large the change was.
  2. The four questions, on the shoe maker's own sales page: relevant expertise in footwear, yes; a track record, unknown; independence from the outcome, no, because the maker profits if you believe the claim; evidence shown, no, because the page names a study and links none. It fails 2 of the 4 questions.
  3. What the failed check shows: the page has offered nothing that should move you either way. The claim itself is still open, so you go to the study.
  4. A budget rebuilt from its inputs (illustrative): a proposal says a math help room during project week costs 600 dollars: 2 tutors × 3 hours × 4 nights × 25 dollars an hour. That is 2 × 3 = 6 tutor hours a night, 6 × 4 = 24 tutor hours, and 24 × 25 = 600 dollars, so the stated total checks out against what it rests on.
  5. A quotation cut short (illustrative): a coach said, "We will not contend this season unless our two injured starters return." A post quotes only "We will not contend this season." The words are accurate, and the meaning flips from a condition to a flat prediction.

Why it matters: the source check asks who is speaking, what they gain, where the claim started, and what the surrounding words say.

Worked example — a flat tire, a garage door, and a memory trial (Week 13)

  1. A failed prediction (illustrative): the hypothesis is that a bike tire goes flat because its valve core leaks. The prediction: a new valve core will stop the leak. After the swap, the tire is flat again by morning. "If the valve core leaks, a new core stops the leak; a new core did not stop it; so the valve core is not the cause." That is modus tollens, a valid form.
  2. A passed prediction: suppose instead the tire holds. "If the valve core leaks, a new core stops the leak; a new core did stop it; so the valve core was leaking." That affirms the consequent: reseating the tire during the swap might have fixed a slow leak at the rim. The pass counts in the hypothesis's favor but cannot settle it.
  3. Rival explanations (illustrative): a garage door reverses halfway down. There are 3 clues: a sensor light blinks, a recycling bin sits beside the sensor, and the remote's light works. There are 3 rivals: a blocked safety sensor, a dead remote battery, and a failed motor. The blocked sensor explains 3 of 3 clues; the dead battery explains none, since the door moves and the remote lights up; the failed motor explains the stop but not the blinking light. The blocked sensor wins on fit, scope, and simplicity, so you test it by moving the bin.
  4. A controlled trial (illustrative): 90 students are randomly assigned, 45 to a memory supplement and 45 to a look-alike placebo, and neither the students nor the testers know who got which. A week later, 27 of 45 improve in each group, and 27 ÷ 45 × 100 = 60 percent in both. The supplement showed no advantage over the look-alike pill.
  5. Testable or not (illustrative): "This charm brings luck, but only to people who truly believe, and doubt cancels it" rules out every possible failure, so it is not testable as stated. "Students who review with flashcards recall more terms a week later than students who reread" could be shown wrong by a trial, so it is testable.

Why it matters: a failed prediction refutes by a valid form, a passed one only supports, and a comparison group is what turns stories into evidence.

Worked example — a cookie rule, a blender tag, and a newsletter deadline (Week 14)

  1. A disconfirming test (illustrative): a café gives a free cookie on some orders. A regular guesses the rule is "orders over 10 dollars" and tests orders of 12 and 15 dollars, both of which get a cookie. Both tests fit the guess and many other rules too, so they could not show it wrong. An order of 8 dollars could: if it also gets a cookie, the guess is wrong.
  2. Anchoring (illustrative): a blender's tag reads "was 80 dollars, now 60." The discount is 80 − 60 = 20 dollars, and 20 ÷ 80 × 100 = 25 percent. The 80 is the anchor; whether 60 dollars is a good price is a separate question you answer by checking other stores.
  3. Overconfidence (illustrative): a club's editors are sure the next newsletter will take 3 days. The last 4 issues took 5, 6, 4, and 5 days. The average is (5 + 6 + 4 + 5) ÷ 4 = 20 ÷ 4 = 5 days, which is 5 − 3 = 2 days over the plan, and none of the 4 finished in 3 days.
  4. Availability (illustrative): after a story about a neighbor's cousin winning a scratch-off prize, a shopper decides the tickets pay off often. How easily one vivid win comes to mind says nothing about how often tickets win; the question is how many tickets win out of how many sold.
  5. The same-standard test (illustrative): a fan dismisses one bad review of a favorite band as "only one critic" and shares one glowing review as proof. Both are single reviews, so the same standard gives both the same weight.

Why it matters: the bias check turns the tests inward: test what could fail, notice the number or story steering you, and hold every piece of evidence to one standard.

Worked example — a shared fridge, two roommates, and a quiet hour (Week 15)

  1. Value and fact (illustrative): a block association argues: P1: Many households on the block run short of fresh food near the end of the month. P2: A laundromat owner has offered a spot by the door for a shared fridge. P3 (supplied): A block ought to set up a shared supply its households need when the space is offered and the upkeep is light. P4: Running the fridge costs about 15 dollars a month in electricity. So, C: The block should set up a shared fridge at the laundromat. P3 says what the block ought to do, so it is the value premise; P1, P2, and P4 are the 3 factual premises.
  2. Which kind of disagreement (illustrative): two roommates agree that chores should be shared fairly. They disagree about whether one of them really cleaned for 6 hours last month. Evidence, such as the chore log, could settle it, so it is a factual disagreement. If they agreed on every hour and still disagreed about whether fair means equal hours or equal tasks, that would be a value disagreement.
  3. The frameworks, side by side (illustrative): a friend weighs whether to keep a promise to help a classmate study, though a party invitation came later. "Keeping it will help my classmate more than the party will help me" is consequence-based. "I promised, and promises bind" is duty-based. "A reliable friend shows up" is character-based. Each framework is a tool for laying out reasons, and this guide gives no verdict.
  4. Steelman and straw man (illustrative): a gym member proposes a nightly quiet hour with no music. The straw man is "they want the gym to be a library." A steelman is "music helps many members push through a hard set, the gym's busiest hour is in the evening, and a quiet hour then would take that help from the most members at once." You reply to the steelman, perhaps by suggesting a quiet hour in the morning.

Why it matters: splitting value from fact shows where a disagreement really lives, and steelmanning makes sure your reply answers the real view.

Worked example — the early tools inside a late question (Weeks 1–8)

A tenants' group bulletin (illustrative) argues: "The neighborhood should open a tool library in the community center's back room. Many renters need a drill or a ladder only a few times a year. The back room sits empty every weekday evening. Stocking 40 basic tools at about 30 dollars each costs about 1,200 dollars."

  1. Argument or explanation (Week 1): the tool library is a proposal still in doubt for the readers, so the passage argues. The opening sentence carries the conclusion with no indicator word, and 3 stated reasons follow it.
  2. Standard form (Week 2): P1: Many renters need a drill or a ladder only a few times a year. P2: The back room sits empty every weekday evening. P3 (supplied): A neighborhood ought to share equipment its residents rarely need when a room is free and the start-up price stays small. P4: Stocking 40 basic tools costs about 1,200 dollars. So, C: The neighborhood should open a tool library in the back room. That is 3 stated premises + 1 supplied premise = 4 premises, and 4 + 1 conclusion = 5 lines. The cost checks: 40 × 30 = 1,200 dollars.
  3. The map (Week 2): P1, P2, and P3 work together to support C, so they are linked; P4 supports P3's "start-up price stays small" condition.
  4. Valid or not (Week 4): a neighbor argues, "If the tool library opens, hardware-store rentals will drop. Rentals dropped. So the tool library opened." That affirms the consequent. Counterexample: a rainy month cut home projects and rentals dropped with no library at all.
  5. The conditional (Week 4): the rule "You may borrow a saw only if you finished the safety demo" means "if you may borrow a saw, you finished the safety demo." The demo is necessary for borrowing but not sufficient, since you also need a library card. The contrapositive, "if you did not finish the demo, you may not borrow a saw," means the same thing.
  6. A Venn verdict (Week 5): "All ladders in the library are inspected. Some inspected items are worn. So some ladders are worn." Shade the part of the ladder circle outside the inspected circle. The X for "some inspected items are worn" can sit in the inspected-and-worn region outside the ladder circle, so no X is forced into the ladder-and-worn overlap. The syllogism is invalid.
  7. Words and relevance (Weeks 6–8): "a cheap drill" is ambiguous, low-priced or poorly made, while "a heavy-duty drill" is vague, with borderline cases. "Ignore the tool-library plan; its organizer has never owned a house" offers a fact about the person, not the plan, so it is an attack on the person (ad hominem). If the back room turned out to be booked every weeknight, P2 would be a false premise, which is a flaw in content, not a fallacy. "Either we open a tool library or every renter buys a drill they use twice" is a false dilemma, since borrowing from a friend is a third option.

Why it matters: the late weeks stand on the early tools: the argument you find, the premise you supply, the form you test, and the reason you check for relevance.

Self-check questions

Try each one, then read the check and the one-line why.

  1. An illustrative claim concerns all 2,500 members of a credit union, and a mailed survey reaches 150 of them. Which group is the sample? Check: the 150 members the survey reaches. Why: the sample is the part actually asked; the 2,500 are the population.
  2. "After the city added a fourth crossing guard, my son started getting to school on time, so the guard is why he is on time." Which fallacy is it? Check: after this, therefore because of this (post hoc). Why: the only support is that one event came first.
  3. An illustrative table shows that towns with more backyard pools report more sunburn cases. Which alternative to a causal reading fits best? Check: a confounding factor, a hot and sunny climate. Why: sunny weather drives both pool building and sunburn, so neither need cause the other.
  4. Support for a bike-share program rose from 20 percent to 25 percent. Report the change both ways. Check: 25 − 20 = 5 percentage points, and (25 − 20) ÷ 20 × 100 = 25 percent. Why: the course reports points and percent together.
  5. A risk rises from 10 in 10,000 people to 15 in 10,000. Give the relative and absolute change. Check: (15 − 10) ÷ 10 × 100 = 50 percent relative, and 15 − 10 = 5 more cases per 10,000 absolute. Why: a relative change needs its base to show how many people it touches.
  6. A product page for a protein bar says "doctors recommend it" and names no doctor. Which credibility question does it fail first? Check: evidence shown. Why: it asserts an endorsement and shows nothing you could check.
  7. A news story quotes a city report; the report is posted on the city's site. Which is the primary source? Check: the city report. Why: it is the original record the story retells.
  8. "Nobody has shown that this mattress does not cure back pain, so it does." Which fallacy is it? Check: the appeal to ignorance (ad ignorantiam). Why: a lack of disproof is offered as proof.
  9. An illustrative trial gives 50 people a new hand cream and nobody a comparison cream. What is missing? Check: a control group, ideally with a placebo cream and blinding. Why: without a comparison, the trial cannot show what would have happened anyway.
  10. A shopper sees "was 50 dollars, now 35" on a lamp. What is the discount, and what is the anchor? Check: 50 − 35 = 15 dollars, and 15 ÷ 50 × 100 = 30 percent; the 50 is the anchor. Why: the old price pulls your sense of value whether or not it was ever fair.
  11. "The town should repaint the faded crosswalks because residents deserve safe streets." Which part is the value premise? Check: "residents deserve safe streets." Why: it says what is owed, not what the situation is.
  12. A classmate says, "Steelmanning the other side means you secretly agree." Fix the claim. Check: steelmanning states the other view at its strongest before replying, and you may still reject it. Why: your reply only counts if it answers the real view.

Check yourself — choose one. Write two or three sentences to a friend.

  • (a) Explain why a valid argument can still have a false conclusion, using an argument you have heard on campus or at work.
  • (b) Explain why a correlation in a table does not by itself show a cause, using a pattern you have noticed in your own life.

Either one rehearses an explain-why skill the final checks.

Watch before the exam

These ten videos come from the course's vetted Week 16 pool, and every one has captions. Use them to review and to connect weeks, not to learn anything new.

Study plan

A short countdown. Count back from the day you plan to sit the final.

  • Two weeks out: as you finish Week 14, start a one-page list of the course's swapped pairs: valid and true, strong and cogent, the converse and the contrapositive, a fallacy and a false premise, a sample and a population, correlation and causation, percentage points and percent, relative and absolute risk, primary and secondary, a confirming and a disconfirming test, and a value premise and a factual premise.
  • One week out: as you finish Week 15, reread the topic reviews above for Weeks 10, 12, and 15. Together they hold 21 of the final's 40 questions.
  • At the start of Week 16: take the practice exam once, cold and closed-note. Note which weeks cost you the most, and remember that it reaches back to Weeks 1–7 far more than the final does.
  • Three or four days before you sit it: reread the topic reviews for your two weakest weeks, and redo the matching worked examples from scratch without looking.
  • Two days before: run the paired exam-prep tutorial (the copy-and-paste chatbot tutor, or its no-AI route) for active recall. Tell it which areas the practice exam exposed.
  • The day before: retake the practice exam, explain every miss out loud, and work the self-check questions above.
  • Right before: skim the "predictable mistakes → cures" lines for Weeks 9 to 15 and the Weeks 1–8 callbacks. Then stop studying; a few slow breaths and a look at your one-page list are enough.

This guide is for review. The exam-prep tutorial and the practice exam are where you rehearse. Pulling ideas back out of memory, not rereading, is what makes them stick.

Exam format

Every one of the 40 questions counts the same, and the final checks your own reasoning without AI. The questions come one at a time, and there is no going back, so settle each one before you move on.

Test-taking strategies for this material:

  • Name the pair before you choose. Most questions turn on two neighboring ideas, such as correlation and causation or a value and a factual premise. Say which pair is in play first.
  • Ask the one-line test. Could the premises be true and the conclusion false? How was the sample chosen? Does the gap survive with the third factor held fixed? Out of how many? Who is speaking, and what do they gain? What observation could show it wrong? Could evidence settle this disagreement?
  • Do the arithmetic on paper. Write each step, then say the number in words before you match it to an option.
  • Treat "illustrative" as a cue. The passage was written for the question and attributed to no one, so judge it by the course's definitions, not by what you guess happened next.
  • For a true-or-false statement, test every part of it. One false clause makes the whole statement false.
  • Refer to an option by what it says. The options are shuffled for each student, so a study partner's "the third one" means nothing on your screen.

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