Lesson 27: WPR II

WPR II Information

WPR II — Today!

Worth: 175 points
Covers: Concepts from Lessons 17–25
Time: 55 minutes
Authorized: Course Statistics Reference Card (SRC) and the issued calculator
Technology: R-Lite (pt, qt, pnorm, qnorm only) — no internet, no electronic devices
Round all numbers to three significant digits

Block II: Inference

Central Limit Theorem (Lesson 17)

If \(X_1, X_2, \ldots, X_n\) are iid with mean \(\mu\) and SD \(\sigma\), then \(\bar{X} \sim N\!\left(\mu, \frac{\sigma^2}{n}\right)\) for large \(n\)
Standard Error: \(\sigma_{\bar{X}} = \frac{\sigma}{\sqrt{n}}\)
Rule of thumb: \(n \geq 30\) unless the population is already normal

Confidence Intervals (Lessons 18–19)

CI for a mean: \(\bar{x} \pm t_{\alpha/2,\, n-1} \cdot \frac{s}{\sqrt{n}}\) (small \(n\)) or \(\bar{x} \pm z_{\alpha/2} \cdot \frac{s}{\sqrt{n}}\) (large \(n\))
CI for a proportion: \(\hat{p} \pm z_{\alpha/2} \cdot \sqrt{\frac{\hat{p}(1-\hat{p})}{n}}\)
Interpretation: the method’s long-run success rate, not the probability a single interval is correct

Hypothesis Testing (Lessons 20–25)

Four steps: Hypotheses \(\to\) Test Statistic \(\to\) \(p\)-value \(\to\) Decision & Conclusion
One-sample \(t\)-test for \(\mu\) (Lesson 21)
One-proportion \(z\)-test for \(p\) (Lesson 22)
Two-sample \(t\)/\(z\)-test for \(\mu_1 - \mu_2\) (Lesson 23)
Paired \(t\)-test for \(\mu_d\) (Lesson 24)
Two-proportion \(z\)-test for \(p_1 - p_2\) (Lesson 25)

Identify the test first: Means or proportions? One sample or two? Paired or independent? Large or small \(n\)?
Check conditions before computing anything — if conditions aren’t met, the test isn’t valid
State hypotheses in terms of population parameters (\(\mu\), \(p\), \(\mu_d\), \(\mu_1 - \mu_2\), \(p_1 - p_2\)) — never \(\bar{x}\) or \(\hat{p}\)
\(H_0\) is always “\(=\)” (status quo); \(H_a\) is what you’re trying to show (\(\neq\), \(<\), or \(>\))
Show your work clearly — partial credit is available
Use proper notation: \(P(X = k)\), \(E[X]\), \(Var(X)\), \(\sigma\)
CI–HT duality: null value outside CI \(\Rightarrow\) reject; inside CI \(\Rightarrow\) fail to reject
Never say “accept \(H_0\)” — only “fail to reject”
Write conclusions in context
Statistical significance \(\neq\) practical significance — always consider effect size

Critical values: qnorm(0.975) for \(z_{0.025}\), qt(0.975, df) for \(t_{0.025}\)

Any questions?