Understanding Sample Size for Proportions
Educational Tool
When planning a study involving proportions (like survey response rates, conversion rates, or success probabilities), you need to determine how many observations are required. This tool helps you calculate sample sizes for confidence interval estimation and hypothesis testing using standard normal approximation formulas.
Single Proportion CI Design
To estimate a proportion with a desired margin of error (ME) and confidence level, you need a sample size that depends on the variability of the proportion. The term p*(1 − p*) represents the variance, maximized at p* = 0.5.
Use p* = 0.5 if you have no prior estimate (conservative approach)
Two Proportions Test Design
To compare two groups with specified alpha (significance level) and power, you need enough observations to detect the expected difference between proportions. Sample size increases rapidly as the effect size shrinks.
Assumes equal group sizes and independent samples
Key Concepts
- Margin of Error (ME): Half-width of confidence interval
- Power (1 − β): Probability of detecting a true effect
- Alpha (α): Significance level (Type I error rate)
- Effect Size: The difference |p₂ − p₁| to detect
- Planning Proportion: Your best guess of the true p
Trade-offs to Consider
- Precision vs. Cost: Smaller ME requires larger n
- Power vs. Resources: Higher power needs more samples
- Effect Size: Smaller effects need much larger samples
- One-sided vs. Two-sided: One-sided tests need fewer samples
- Confidence vs. Uncertainty: Higher confidence = wider intervals
Assumptions & Limitations
- Normal approximation: These formulas assume the sample proportion is approximately normal. This works well for np ≥ 10 and n(1−p) ≥ 10.
- Simple random sampling: Each observation is independent and has equal probability of selection. Complex surveys may need different methods.
- Equal group sizes: Two-sample calculations assume equal n per group. Unequal allocation requires different formulas.
- Educational purposes only: Do NOT use for clinical trials, regulatory submissions, or high-stakes decisions without proper statistical consultation and more rigorous analysis.
- Rounding effects: Sample sizes are rounded up, so achieved precision or power may slightly exceed targets.
Frequently Asked Questions
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