Sample size determination is the process of deciding how many observations are needed before collecting data. It matters because too small a sample can give results that are too uncertain, while too large a sample can waste time, money, or participants. In estimation problems, the goal is often to achieve a chosen margin of error at a chosen confidence level.
In hypothesis tests or experiments, the goal is often to have enough power to detect an important effect.
Key Facts
- For estimating a population mean with known or planned standard deviation: n = (z*σ / E)^2
- For estimating a population proportion: n = z*^2 p(1 - p) / E^2
- If p is unknown for a proportion, use p = 0.50 because it gives the largest required sample size.
- Margin of error for a mean is E = z*σ / sqrt(n), so quadrupling n cuts E in half.
- For comparing two equal-size group means, a planning formula is n per group = 2(zα/2 + zβ)^2 σ^2 / Δ^2
- Power is the probability of detecting a real effect, commonly set to 0.80 or 0.90 before the study begins.
Vocabulary
- Sample size
- The number of observations, measurements, or participants included in a study.
- Margin of error
- The maximum expected distance between a sample estimate and the true population value for a stated confidence level.
- Confidence level
- The long-run percentage of confidence intervals made by the same method that would contain the true population value.
- Statistical power
- The probability that a test correctly rejects a false null hypothesis when a real effect exists.
- Effect size
- The size of the difference or relationship a study is designed to detect, often written as Δ for a difference in means.
Common Mistakes to Avoid
- Using n instead of sqrt(n) in the margin of error formula is wrong because uncertainty decreases with the square root of sample size, not directly with sample size.
- Forgetting to round the final sample size up is wrong because a fraction of an observation cannot be collected and rounding down may miss the target precision or power.
- Using p = 0.50 only after data are collected is wrong because p = 0.50 is a conservative planning value used before the true population proportion is known.
- Confusing confidence level with power is wrong because confidence describes interval coverage for estimation, while power describes the chance of detecting a real effect in a test.
Practice Questions
- 1 A researcher wants to estimate a population mean with σ = 12, margin of error E = 3, and 95% confidence. Use z* = 1.96 to find the required sample size, rounding up.
- 2 A pollster wants a 95% confidence interval for a population proportion with margin of error E = 0.04 and no prior estimate of p. Use p = 0.50 and z* = 1.96 to find the required sample size, rounding up.
- 3 A study team can either increase confidence from 95% to 99% or decrease the desired margin of error while keeping the same population variability. Explain how each choice affects the required sample size and why.