Activity 32
MATH 216: Statistical Thinking
Power Analysis & Type II Error
Time Allocation: 15 minutes total (5 min reading, 10 min individual work)
Part 1: Conceptual Understanding (3 minutes)
Instructions: Answer the following questions:
- What is Type II error (\(\beta\)) and how does it relate to statistical power?
- How does increasing the effect size (distance between \(\mu\) and \(\mu_0\)) affect \(\beta\) and power?
- What factors influence statistical power besides effect size?
Part 2: Power Calculations (12 minutes)
Perform power analysis calculations for the following 4 scenarios:
Example 1: Medical Testing
Context: Diagnostic test for disease detection
- \(H_0\): Patient is healthy (no disease present)
- \(H_a\): Patient has disease (medical condition present)
- \(\alpha = 0.05\), \(\beta = 0.20\)
Calculate: Power =
Example 2: Quality Control
Context: Manufacturing defect rate monitoring
- \(H_0\): \(p = 0.05\) (acceptable defect rate)
- \(H_a\): \(p > 0.05\) (excessive defects)
- \(n = 100\), \(\alpha = 0.05\), \(p_{\text{actual}} = 0.08\)
Calculate:
- Rejection boundary =
- \(\beta\) =
- Power =
Example 3: Battery Life Study
Context: Testing new battery technology
- \(H_0\): \(\mu = 2400\) hours
- \(H_a\): \(\mu > 2400\) hours
- SE = 28.27, \(\alpha = 0.05\), critical \(\bar{x} = 2446.5\)
- For \(\mu = 2450\): \(\beta\) = , Power =
- For \(\mu = 2475\): \(\beta\) = , Power =
Example 4: Rat Response Times
Context: Neuroscience experiment on reaction times
- \(H_0\): \(\mu = 1.2\) seconds
- \(H_a\): \(\mu \neq 1.2\) seconds
- SE = 0.05, \(\alpha = 0.01\), critical values: \(\bar{x} = 1.329, 1.071\)
- For \(\mu = 1.1\): \(\beta\) = , Power =
- For \(\mu = 1.25\): \(\beta\) = , Power =
Show your work for one calculation: