Exam 2 Study Guide

Modules 6–9 · Probability, estimation, and hypothesis testing

This guide tells you what Exam 2 asks you to do, and where on the site each skill is taught. It doesn’t contain new examples. Every item below links to a Module section, a pre-study activity, or a lab step you’ve already worked through, so you can go straight to the material that matters and check that you can do it.

How to use it. Cover the Where links, read each Ready if you can statement, and try to explain or demonstrate the skill without looking anything up. If you get stuck, the links tell you exactly where to practice.

What Exam 2 looks like

	Part A · written	Part B · coding
When	Monday, in class	Wednesday, in class
Time	Up to 90 minutes; most students need about 45 to 60	Up to 75 minutes; most students need about 45
Points	70	30
You may use	One hand-written, letter-size (8½ × 11 inch) sheet of your own notes (both sides), a standard (basic) calculator, and a pencil. No other notes, no phone or laptop.	The course website and your own hand-written notes. No other websites, and no AI assistance.
What you do	Reason about the ideas: work short calculations by hand, read printed R output, choose a test, and interpret results. There is no code to write, and every critical value you need is printed in the question.	Run a series of analysis tasks on an exam webR page, like the pre-study pages, and record your results on a paper answer sheet.
How you answer	Write a number in a box, mark one option, or mark one box per row of a grid. One question asks for a short written explanation.	Write numbers in boxes, mark a few options, and write one APA-style sentence.

Part A’s points are spread across four sections: probability (M06), estimation (M07), testing (M08), and choosing and running tests (M09). Testing and choosing tests carry the most weight.

What’s in scope. Everything on Exam 2 is taught in the main content of Modules 6–9, their pre-study pages, and their labs. The indigo Going further boxes, the bonus pre-study pages, and the bonus labs aren’t tested; as How to Read This Site says, nothing in them is required. Part B also uses the M04 and M05 skills you’ve been using all semester.

Part A · what to be ready for

Because Part A is closed to the website, try each Ready if you can check with the website closed. If you can’t do it yet, follow the links.

Probability (M06)

1. Probabilities from a table. Read a marginal probability, compute a conditional probability, and check whether two events are independent by comparing their joint probability with the product of their separate probabilities. Where: Three types of probability from a table · Conditional probability · The multiplication rule revisited · Pre-study Activities 2.2 and 2.4 · Lab Step 3

Ready if you can: given a table of joint probabilities or counts, find P(A), P(A | B), and the value P(A and B) would take if A and B were independent, then decide whether they are.

2. The binomial distribution. The expected count, a cumulative probability read from a printed table, and what that probability does and doesn’t describe. Where: Bernoulli trials · Calculating binomial probabilities · Expected value · Pre-study Activities 4.1 and 4.2

Ready if you can: compute np, add up the probabilities for “k or more” from a printed table, and say in words what that probability is conditional on.

3. Core ideas, in words. Probability as area for a continuous variable; what the central limit theorem and the law of large numbers say (and what they don’t); parameters versus statistics. Where: Probability as area · What happens as n grows · Parameters vs statistics · M06 core concepts · M07 · the central limit theorem

Ready if you can: say which symbols name population values and which name sample values, and explain exactly what the central limit theorem says becomes approximately normal.

Estimation (M07)

4. A confidence interval for a mean, by hand. The standard error, then the mean plus or minus \(t^*\) times the SE. Where: The SE formula · Building the CI · Pre-study Activity 3.2

Ready if you can: from n, the mean, the SD, and a given \(t^*\), compute the SE and both ends of the interval, and say how the SE changes when n is four times larger.

5. What “95% confident” means. The long-run success rate of the method, not a probability about one interval, and not a range for individual scores. Where: What 95% confident actually means · The 95% is a choice · Pre-study Video 1

Ready if you can: sort statements about an interval into fine and wrong, and say how changing the confidence level changes the width.

6. Confidence interval versus prediction interval. Where: CIs vs PIs · Quick decision rule · Why PIs are wider · Pre-study Activities 4.1 and 4.2

Ready if you can: decide whether a goal calls for a CI or a PI, and explain why the PI is wider.

Testing (M08)

7. The one-sample t-test, start to finish. Hypotheses written about the parameter, \(t_\text{obs}\), the degrees of freedom, the decision against a given \(t^*\), and Cohen’s d. Where: The null hypothesis · The four ingredients · Compute \(t_\text{obs}\) · The critical value · Cohen’s d, one sample · Pre-study Activities 1.1 and 4.2

Ready if you can: pick the correct pair of hypotheses (and say why hypotheses are about μ, not x̄), compute \(t_\text{obs}\), df, and d from summary numbers, and make the decision.

8. Intervals and tests agree. For a matching two-sided test at α = .05 and a 95% confidence interval, the test rejects the null hypothesis exactly when the hypothesized value falls outside the interval. Where: Three readings of one calculation · The CI–NHST connection

Ready if you can: state the decision from an interval alone, and check that it matches the test’s decision.

9. The null distribution and the logic of the test. Why the reference distribution is centered where it is, which areas make up a two-sided p-value, and why a test starts by assuming the skeptic is right. Where: The null sampling distribution · The p-value · The rejection region · Pre-study Activity 2.4 and Video 3

Ready if you can: shade the two-sided p-value on a sketch of a t distribution, and explain in three or four sentences why the test assumes the null hypothesis and how the reference distribution lets you judge the result.

10. What a p-value is, and isn’t. Where: The p-value · Setting α in advance · A field guide to misconceptions · Pre-study self-test

Ready if you can: mark statements about a p-value true or false, including ones about the probability of the null, about practical importance, about replication, and about a different α.

11. Type I and Type II errors, and power. Where: Type I and Type II errors · Statistical power · The fundamental tradeoff

Ready if you can: classify each combination of “what’s true” and “what the study concludes” as a Type I error, a Type II error, or a correct decision; say what a power value means in words; and say whether power goes up or down when the sample size, α, the true effect, or the variability changes.

Choosing and running tests (M09)

12. Choosing the test. One-sample t, two-sample t, one-way ANOVA, chi-square goodness of fit, or chi-square test of independence. Where: The question comes before the test · The decision map · Goodness of fit · The tests side by side

Ready if you can: for a research question, name the outcome type, the number of groups, and whether it compares to a fixed value, and from those pick the test.

13. Reading two-sample t output. Telling the standard and Welch versions apart, and why t and Cohen’s d answer different questions. Where: Run and read the test · Quantify the effect · Pre-study Activities 2.2, 2.3, and 3.2

Ready if you can: look at two rows of t_test() output and say which is Welch’s and how you know, and explain why t can be large while d is moderate.

14. Where a causal claim comes from. Random assignment, not the test. Where: Where does the causal claim come from? · What’s new in Test 1

Ready if you can: say what licenses a causal sentence in a randomized study and what the test itself tells you.

15. The one-way ANOVA table. Mean squares, F, the null hypothesis, what a rejection does and doesn’t tell you, and why running several unadjusted t-tests is a problem. Where: Test 2 · Recognize the design · Two kinds of variation · Run and read the test · Pre-study Activity 4.1

Ready if you can: fill in missing mean squares and F in a printed ANOVA table, state H₀ about the population means, write the conclusion a rejection supports, explain why running several unadjusted pairwise t-tests increases the chance of at least one Type I error, and say what an omnibus ANOVA can — and cannot — tell you about differences among the group means.

16. The chi-square test of independence, by hand. Expected counts, one cell’s contribution, the total, the degrees of freedom, the expected-count condition, and the conclusion. Where: Test 3 · Assess the model · Run and read the test · Pre-study Activity 5.2

Ready if you can: from a table of counts with totals, compute an expected count, (O − E)²/E for a cell, and the full χ², give its df, check the expected-count condition, and pick a conclusion that doesn’t overstate the result.

Part B · what to be ready for

Part B is a sequence of tasks on one dataset, each building on the skills from the labs and pre-study. The website is open, so the skill being tested is knowing which recipe to use and where it lives, then running it carefully. For each task type below, find the linked code now and make sure you can run it from a blank code box.

1. Create a new variable, then summarize it by group. Where: M04 · mutate() · M04 · group_by() and summarize() · M05 · group summaries

Ready if you can: add a new column with mutate() and save it back into the data, then, as a separate step, produce each group’s n, mean, and SD. Keep the two steps apart: if you assign a summary table back to the dataset’s name, the dataset becomes the summary and later tasks can’t find the new column.

2. A bootstrap percentile interval for one group’s mean. Where: M07 · the bootstrap · Constructing the percentile CI · Pre-study Activities 2.2, 2.3, and 2.4

Ready if you can: filter to one group and calculate its observed mean; set the seed immediately before resampling; use rep_slice_sample() to generate 1,000 bootstrap samples, as M07 does (each the group’s full size, sampled with replacement); calculate one mean per resample; and report the number of bootstrap means, their standard deviation (the bootstrap SE), and the 2.5th and 97.5th percentiles. Keep the observed mean and the bootstrap means distinct: the first comes from the data, the second set from the resamples.

3. A two-sample t-test, Cohen’s d, and an APA sentence. Where: M09 · run and read Test 1 · Quantify the effect · APA reporting, two-sample t · Pre-study Activities 2.3 and 3.1 · Lab: the t-test, Cohen’s d, and the APA line

Ready if you can: run whichever version of the test a task specifies (standard or Welch), with the groups in the order it asks for; read off the difference, its interval, t, df, and p; compute the matching d and check that its sign agrees with t; and write one APA sentence with all of them.

4. A one-way ANOVA, η², and Tukey’s comparisons. Where: M09 · run and read Test 2 · Quantify the effect · APA reporting, ANOVA

Ready if you can: fit the model, build the ANOVA table, and read F, both df, and p; compute η²; and get Tukey’s adjusted p-value for each pair of groups.

5. A chi-square test of independence and Cramér’s V, and what a nonsignificant result means. Where: M09 · run and read Test 3 · Quantify the effect · M08 · the language of decisions · M09 · where the causal claim comes from

Ready if you can: run the test without the continuity correction, compute Cramér’s V, and say correctly what “fail to reject” does and doesn’t let you conclude, including what random assignment does and doesn’t guarantee about the groups.

Part B habits that save points

Do the tasks in order. Later tasks use what earlier ones create.
Check before you record. Glance at each result and ask whether it makes sense: is the sign what you expect, is the n right?
Record numbers the way the sheet asks, usually two decimals, and p-values as decimals or “< .001.”
If a task won’t run, record any results you already have, leave the remaining boxes blank, and move on. Return to it if time permits. Do Task 1 first: later tasks use the variable it creates.

Making your one-page sheet for Part A

You may bring one hand-written, letter-size (8½ × 11 inch) sheet of your own notes, both sides. Building it is good studying in itself: deciding what goes on it makes you sort out what you understand from what you’d need to look up. Ideas worth including:

The probability rules and how to check independence.
The binomial expected value.
The standard error, the confidence interval, and how the SE changes with n.
\(t_\text{obs}\) for one sample, and Cohen’s d for one and two samples.
The degrees of freedom for each test.
For the ANOVA table, mean square = sum of squares ÷ df, and F = between ÷ within.
For chi-square, the expected count (row total × column total ÷ grand total), a cell’s contribution, and the expected-count condition.
One-line interpretations you trust, for a 95% interval, for a p-value, and for power.

You don’t need a table of critical values; every one you need is printed in the question.

The week before the exam

A few days out: work through the Part A checks above with the website closed, and mark any you can’t do yet. Follow their links and try again later the same day.
Redo pre-study activities from the ✍️ tab only. Don’t open 💡 Hint or 👀 Spoiler until you’ve written your own answer.
For Part B, open a pre-study page and rerun each recipe above from a blank code box, with your lab notebooks beside you. Practice finding the right section of the site quickly.
The day before: finish your notes sheet, and reread the Summary of M06, M07, M08, and M09.
Questions? Ask before the exam. A question asked is always better than a guess.

Bring: on Monday, a pencil, a standard (basic) calculator, and your notes sheet; on Wednesday, your laptop (charged) and a pencil for the answer sheet. The syllabus has the dates.