Week 12 — Review & Exam 2 (Ch 4–5)

This week in a line: no new material — this is your Chapters 4–5 review and worked-example study guide before Exam 2.

Exam 1 asked you to describe what was already sitting in front of you — a batch of numbers, a two-way table, a bell curve you were simply handed. Chapters 4 and 5 ask a harder, more honest question: what can you say about a population you cannot fully see, using only a sample drawn from it? Chapter 4 sharpened the Normal curve from Chapter 3 into an exact instrument — any cutoff, any percentile, no more rounding to the nearest whole standard deviation. Chapter 5 then turned that instrument outward: every sample statistic ($\hat p$, $\bar x$) wobbles a little from sample to sample, and Chapter 5 gave you the vocabulary and formulas — standard error, confidence interval, hypothesis test — to say exactly how much wobble to expect and what to conclude from it. Exam 2 checks whether you can run both halves cold, in whichever order a question demands. This unit is your self-study companion: a review checklist by chapter, the new formula sheet for Ch 4–5, seven fully worked exam-style problems, and a short list of exam-day tips. There is no new content this week — if a term below feels unfamiliar, go back to the week that introduced it before you test.

1Review checklist: what “ready for Exam 2” looks like¶

Work down this list chapter by chapter. For each bullet, ask yourself: could I do this without any notes or preparation, right now? If not, redo that week’s worked example without looking at the solution first.

1.1Chapter 4 — Working with the Normal Model (Week 8)¶

• Standardize any value with the z-score $z=(x-\mu)/\sigma$ and read off the exact proportion below, above, or between two cutoffs using normalcdf (TI) / xpnorm (R) — no more rounding to the nearest whole standard deviation.

• Work the inverse-Normal direction with invNorm (TI) / xqnorm (R): given a percentile, or a “top $p$%” / “bottom $p$%” statement, find the cutoff value.

• Translate “at least” (right tail) and “at most” (left tail) into the correct shaded region before you compute anything, and know the -1E99 / 1E99 open-tail convention on the TI.

• Reconnect the exact method to the empirical rule from Chapter 3: know that “95% within $\mu\pm2\sigma$” is a rounded shortcut for the more exact figures normalcdf/xpnorm return at $z=\pm2$ (about 95.4%), and that the empirical rule is a close approximation, not a replacement, for the exact calculation.

• Judge whether a real distribution is close enough to bell-shaped to trust the Normal model (mean close to median, no long one-sided tail), and name a variable for which it would be a poor choice (e.g., household income).

1.2Chapter 5 — Foundations for Inference (Weeks 9–11)¶

• Tell a population parameter ($\mu$, $p$) apart from a sample point estimate ($\bar x$, $\hat p$), and explain that every point estimate carries sampling variability — a different sample would produce a slightly different number.

• Compute the standard error of a proportion, $SE_{\hat p}=\sqrt{\hat p(1-\hat p)/n}$, and of a mean, $SE_{\bar x}=s/\sqrt{n}$ — the SE is the typical size of that sample-to-sample wobble.

• State the Central Limit Theorem in plain language: for a large enough sample, the sampling distribution of $\bar x$ (or $\hat p$) is approximately Normal, centered at the true parameter, with spread equal to the SE.

• Build a large-sample confidence interval as point estimate $\pm\, z^*\cdot SE$, and interpret it correctly — “we are 95% confident the true [parameter] is between ___ and ___” — never “there is a 95% probability the parameter itself falls in this interval.”

• Explain how confidence level and sample size each change the margin of error: higher confidence widens the interval; a larger $n$ narrows it. Know the three critical values by heart: $z^*=1.645$ (90%), 1.96 (95%), 2.576 (99%).

• Write a null hypothesis $H_0$ and alternative hypothesis $H_a$ for a one- or two-sided claim, compute a test statistic $z=(\text{estimate}-\text{null value})/SE$, and use the p-value to reach a decide / fail-to-reject conclusion at a stated significance level $\alpha$.

• Distinguish a Type I error (rejecting a true $H_0$) from a Type II error (failing to reject a false $H_0$), and state the real-world cost of each in a given scenario.

• Distinguish statistical significance (a small p-value) from practical importance (whether the difference is actually large enough to matter in context).

2What belongs on your formula sheet¶

Your formula sheet may hold formulas, symbol definitions, and short notes — not worked examples. Here is the complete list for Ch 4–5.

Critical values $z^*$ (memorize — short enough to skip the sheet):

Symbol legend to have memorized (notation, not formulas, so it doesn’t belong on the sheet): $\mu$ (population mean) vs. $\bar x$ (sample mean); $p$ (population proportion) vs. $\hat p$ (sample proportion); $H_0$ (null hypothesis) vs. $H_a$ (alternative hypothesis); $\alpha$ (significance level, chosen before you see the data); $n$ (sample size).

3Exam-style worked examples¶

Work each one yourself first — cover the solution with your hand or a sheet of paper — then check your steps against what’s shown. Every number below was computed in R (mosaic/BSDA, R 4.5.2, set.seed(1209)) so you can trust it to the decimal place.

4Quick self-check before the exam¶

Try each one on paper first. Brief answers follow so you can check yourself — no full step-by-step solutions are given here on purpose; if your answer doesn’t match, redo the matching worked example above before you test.

1. Using Example 1’s playlist model $N(3.5, 0.6)$, what proportion of songs run over 4.5 minutes? (Same tail as the “over 60 min” idea in Example 7(a): $z=1.667$, $P(X>4.5)=1-0.9522=0.0478$ — about 4.8%.)

2. In Example 2, if the challenge instead wanted the top 5% cutoff (an “elite” badge), would that step-count value be higher or lower than the 9,715-step “top 20%” cutoff found above, and why? (Higher — a smaller percent at the top means a farther-out, more extreme cutoff.)

3. In Example 3, if the water-bottle sample size had been $n=600$ instead of $n=150$ (same $\hat p=0.28$), would $SE_{\hat p}$ get bigger or smaller, and by roughly what factor? (Smaller; $n$ is under a square root, so quadrupling $n$ cuts the SE in half.)

4. In Example 4’s carpool interval $(0.2085, 0.3315)$, is it correct to say “there’s a 95% chance the true carpool proportion is between 20.9% and 33.2%”? Why or why not? (No — the true proportion is fixed; the 95% describes how often the interval-building procedure captures it across repeated samples, not a probability about this one fixed interval.)

5. In Example 5, would a 99% confidence interval for mean sleep be wider or narrower than the 90% interval $(6.535, 6.865)$ shown? (Wider — higher confidence requires a bigger margin of error, i.e., a bigger $z^*$, 2.576 vs. 1.645.)

6. In Example 6’s laptop test, if the sample had instead found exactly 162 owners out of 180 ($\hat p = 0.90$, matching the claim exactly), what would the test statistic $z$ equal, and what decision would follow? ($z=0$; p-value $=1$; fail to reject $H_0$ — no evidence against the claim.)

7. What is the difference between a Type I and a Type II error, in one sentence each? (Type I: rejecting a true $H_0$. Type II: failing to reject a false $H_0$.)

8. In Example 7, which of the three scenarios would change from a hypothesis test to a confidence interval if the facilities manager instead asked “what’s our best estimate of the true average fill amount?” rather than “does it differ from the claimed 12 oz?” (Scenario (c) — same data, different goal: estimating vs. testing a specific claim.)