Week 15 — Final review: What claim can we responsibly make?

MATH 21003 · Introduction to Statistical Methods · Fall 2026 · Week 15 (Mon Dec 7, 2026 — last class meeting)

This is the last class meeting

This week is a single meeting: Monday, December 7, 2026. There is no Wednesday and no Friday class. There is no Week 15 exit ticket, no quiz, and no new homework. The cumulative final exam is given during the university’s final-exam window (December 9–15); the exact date, time, and room are posted in Blackboard, not here.

So this page is not a new lesson. It is the final review — one walkthrough that pulls the whole term back together around the question the course has been circling since Week 1.

The closing question

Every week, under the vocabulary and the graphs, you were really learning to answer one question:

What claim can we responsibly make?

Not “what does the data say it wants to say,” and not “what would make the best headline” — but what can we honestly conclude, and just as importantly, what can we not? A responsible claim is one that matches its evidence: it names the kind of study it came from, it reports the size and the uncertainty of the effect, and it stops where the evidence stops. The final exam is, at bottom, a test of whether you can do that on a scenario you have not seen.

The arc of the course, in one read

You do not need to re-study every week as if it were new. It is more useful to see how the weeks fit into a single chain of reasoning. Read it once, straight through.

Data and variables (Week 1). Every analysis starts by naming the cases (who or what is in a row) and the variables (numerical or categorical; response vs explanatory). If you cannot say what the cases and variables are, you cannot yet say anything responsible.

Study design and causation (Weeks 2, 6). Where did the data come from? An experiment with random assignment can license a causal claim; an observational study, by itself, usually cannot. Confounding — a third variable tied to both the explanatory variable and the response — is the standing reason an observational association can mislead. The honest move is the alone / after accounting for sentence: what the relationship looks like on its own, and what changes once you adjust for the obvious confounder.

Visualization and summaries (Weeks 3, 4). Describe one variable by its center, spread, shape, and unusual values; compare groups with a difference in means or proportions. A numerical difference is real, but whether it matters and whether it is causal are separate questions.

Association (Week 5). A scatterplot and a correlation describe how two numerical variables move together — the linear part of it. Correlation is a description, not a cause, and a small r means no apparent linear trend, not “no relationship.”

Sampling and variability (Weeks 10, 11). A sample is not the population, and a statistic computed from one sample would land somewhere slightly different in the next. That sampling variability is exactly why we need intervals and tests — to separate a real signal from the noise of “which sample did we happen to get.” Week 10 also reframed probability as risk, and showed how base rates change what a positive test result means.

Confidence intervals (Weeks 11–12). A confidence interval is a range of plausible values for the thing we are trying to estimate. A wide interval means an imprecise estimate; a narrow one means a precise estimate. The interpretive habit: an interval that includes the no-effect value (0 for a difference, 1 for a ratio) cannot rule out “no effect.”

p-values and hypothesis tests (Weeks 11–12). A p-value measures how surprising the data would be if nothing were going on — strength of evidence, not a verdict. Simulation-based reasoning (Week 11) and the classical t/z/χ² tests (Week 12) are two routes to the same idea. Keep the two error types straight, and resist the ritual of “reject / fail to reject” as if it were the whole story.

Categorical outcomes — RR and OR (Week 13). For yes/no outcomes in a \(2\times2\) table: the risk difference, the relative risk (RR), and the odds ratio (OR). RR and OR are ratios, so their no-effect value is 1. An OR is not a probability, and a large relative risk can still be a small absolute risk.

Meta-analysis and forest plots (Week 14) — short callback. When many studies address one question, a meta-analysis combines their estimates into a single pooled estimate, pictured as a forest plot (each study a row with its interval; the pooled result a diamond). The grown-up caution: pooling cannot rescue biased or mismatched studies — a precise diamond is still not proof. That is as far as we take meta-analysis; the methods behind the pooling are beyond this course.

Practical vs statistical significance. A result can be statistically discernible — the interval clears the no-effect value, the p-value is small — and still be too small to matter in practice. “Detectable” and “important” are different questions, and a responsible claim answers both.

Evidence strength and limitations. The thread that runs through all of it: no single number is the whole story. Design limits what you can claim; confounding can fake an association; precision is not the same as validity; and heterogeneous or publication-biased evidence can be confidently wrong. The conclusion always carries its limits with it.

A responsible-claim checklist

When a scenario lands in front of you — on the final, or in real life — walk these questions in order. They are the same questions, in the same order, the whole course practiced.

What is the question? State plainly what is being asked.
What are the variables and groups? Cases; response vs explanatory; numerical vs categorical; which groups are being compared.
What was the study design? Experiment or observational? Random assignment? What is the scope of inference (who and when)?
What estimate or comparison was used? A difference in means or proportions, a correlation, a slope, a risk ratio or odds ratio, a pooled estimate?
What does the interval or test say? Does the confidence interval include the no-effect value (0 or 1)? Is the p-value strong or weak evidence?
Is the effect practically meaningful? Detectable is not the same as important — is the size of the effect enough to matter?
What can and cannot be concluded? Write the one-sentence conclusion the evidence supports — and name what it does not support.
What bias, confounding, or heterogeneity issues matter? Name the most plausible threat to the claim, and say which direction it could push the result.

A good answer on the final is rarely a single number. It is a short, honest paragraph that moves through this list.

One scenario, read end to end

Here is the kind of integrated reading the final rewards. It reuses a familiar, illustrative scenario from the course (the same small blood-pressure trial we used at the midterm) — not a real study, just a clean case to practice the whole chain on. No new data, no calculator gymnastics; the work is in the reasoning.

A small trial randomly assigns 200 adults with high blood pressure: half to a new medication, half to a placebo. After eight weeks, the medication group’s average systolic blood pressure is about 138 mm Hg and the placebo group’s is about 145 mm Hg — a difference of about 7 mm Hg in favor of the medication. Suppose a 95% confidence interval for that 7 mm Hg difference does not include 0.

Now walk the checklist.

Question. Does the new medication lower systolic blood pressure compared with placebo?
Variables and groups. Cases are the 200 adults; the response is systolic blood pressure (numerical); the explanatory variable is treatment group (medication vs placebo — categorical). The comparison is a difference in means (Week 4).
Design. A randomized experiment. Because participants were randomly assigned, the comparison is a fair causal one (Week 2): random assignment is what lets us move from “the medication group was lower” to “the medication appears to lower blood pressure.”
Estimate and uncertainty. The estimate is the 7 mm Hg difference; the confidence interval is its range of plausible values (Weeks 11–12). Since the interval excludes 0, the data are evidence of a real, non-zero effect — a small p-value would say the same thing.
Practically meaningful? This is the separate question. A 7 mm Hg drop is clinically modest but not trivial; whether it is “enough” depends on the patient, the side effects, and the alternatives. Detectable and important are not the same claim.
Conclusion, with limits. In this randomized trial, the medication lowered average systolic blood pressure by about 7 mm Hg compared with placebo, and the data are consistent with a real effect. What it does not claim: that the effect is large, that it holds for people unlike these 200 adults, or that one trial is the last word.
Threats. Random assignment handles confounding here (its main job). The open limits are the small, specific sample (scope of inference) and the fact that this is one study — which is exactly the door Week 14 opened: the responsible next step is to ask what all the trials of this medication show together, not just this one.

That is the whole course on one case: design, comparison, inference, the practical-vs-statistical distinction, and a conclusion that names its limits.

Studying for the final

The final exam is cumulative and case-based — short scenarios with graphs, tables, or output, and written-conclusion questions — not a formula dump. To prepare:

Review by re-reading, not re-deriving. Skim each week’s note page and ask, for each, what claim does this tool license, and what does it not? The weekly “What you should be able to do” lists are a fast self-check.
Practice the checklist out loud. Take any scenario from a past exit ticket, quiz, or note page and walk the eight questions above. The exam rewards the order of reasoning, not memorized definitions.
Rehearse the cautions. Association vs causation; confounding; “detectable ≠ important”; an interval that includes the no-effect value; RR/OR no-effect value of 1; pooled ≠ proof. These are the sentences that turn a half-answer into a responsible one.
Write short, honest conclusions. Practice ending every scenario with one sentence the evidence supports and one phrase about what it does not.

You will not find exam questions on this page, and you should not try to predict them. Practice the thinking and the specific items take care of themselves.

Where things live

🔒 Final exam. Cumulative; given in the Dec 9–15 university window. The exact date, time, room, and any rules (calculator, formula sheet) are posted in Blackboard. This page does not contain exam content.
🔒 Project. Your project deliverable and its due date are handled in Blackboard; the submission window was earlier in the closing stretch. Check Blackboard for your current status.
Office hours and the consultation day (Tue Dec 8). Use them — bring a scenario you found hard and walk it through the checklist with me.
No new homework, exit ticket, or quiz this week.

This review opens no new chapters. To brush up a specific idea, go back to the week where the course first built it — the weekly note pages are the review readings, and the OpenIntro books behind them (IMS and ISLBS, both CC BY-SA 3.0) remain the second voices for any topic you want to see again:

Design and data: Weeks 1–2, 6.
Description and association: Weeks 3–5.
Probability, risk, and inference: Weeks 10–12.
Categorical outcomes and evidence synthesis: Weeks 13–14.

About this page: This is a final-review synthesis page. It introduces no new statistical method; it reviews material the course developed across Weeks 1–14 from OpenIntro Introduction to Modern Statistics (Çetinkaya-Rundel & Hardin) and OpenIntro Introductory Statistics for the Life and Biomedical Sciences (Vu & Harrington), both shared under CC BY-SA 3.0. The blood-pressure scenario is an illustrative teaching example reused from the course, not a real study. Course materials by Matt Hester, shared under CC BY-SA 3.0.

Reuse

CC BY-SA 3.0.