Activity 32

MATH 216: Statistical Thinking

Activity 1: Think-Pair-Share

When the true mean \(\mu\) moves farther from the null hypothesis value \(\mu_0\), how does \(\beta\) (Type II error probability) change? Explain why this occurs, considering the sampling distributions under \(H_0\) and \(H_a\).

Activity 2: Calculate Power

For \(H_0: \mu = 2400\) vs. \(H_a: \mu > 2400\), the rejection region is \(z > 1.645\). If the true \(\mu = 2450\) and the critical value in original units is \(\bar{x} = 2446.5\), calculate \(\beta\). Assume \(\sigma/\sqrt{n} = 28.27\). How does \(\beta\) compare to the case where \(\mu = 2425\)?

Activity 3: Data Exploration

A dataset shows drug-injected rat response times (in seconds). The control mean is \(\mu_0 = 1.2\). Use visualizations and summary statistics to explore whether the drug-injected rats differ from \(\mu_0\).

Summary Statistics: \(\bar{x} = 1.15\), \(s = 0.5\), \(n=100\)

Activity 4: Group Power Analysis

For the rat response time data (\(\mu_0 = 1.2\), \(\alpha = 0.01\)), calculate \(\beta\) and power (\(1 - \beta\)) for \(\mu = 1.1, 1.15, 1.25\).

Activity 5: Case Study

A neurologist tests if a drug changes rat response times (\(\mu_0 = 1.2\), \(\alpha = 0.01\)). For \(\mu = 1.1\), calculate \(\beta\) and power. Discuss implications.