Power Analysis & Sample Size Reference Cheat Sheet

A printable reference covering power, significance level, effect size, sample size, confidence intervals, and error types for grades 11-12.

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Power analysis helps students plan studies before collecting data by estimating how large a sample is needed to detect a meaningful effect. This cheat sheet connects hypothesis testing, confidence intervals, effect size, and sample size in one reference. It is useful for designing experiments, evaluating survey claims, and understanding why small studies often miss real effects.

Key Facts

Statistical power is the probability of correctly rejecting a false null hypothesis, so $\text{Power} = 1 - \beta$ .
A Type I error occurs when a true null hypothesis is rejected, and its probability is $\alpha$ .
A Type II error occurs when a false null hypothesis is not rejected, and its probability is $\beta$ .
For estimating a population mean with margin of error $E$ , an approximate sample size is $n = \left(\frac{z_{\alpha/2}\sigma}{E}\right)^2$ .
For estimating a population proportion with margin of error $E$ , an approximate sample size is $n = \frac{z_{\alpha/2}^2 p(1-p)}{E^2}$ .
When $p$ is unknown for a proportion sample size calculation, use $p = 0.5$ because it gives the largest required $n$ .
Cohen's standardized mean effect size is $d = \frac{\mu_1 - \mu_2}{\sigma}$ .
Increasing $n$ , increasing effect size, increasing $\alpha$ , or reducing variability generally increases statistical power.

Vocabulary

Power: Power is the probability that a test detects a real effect when the alternative hypothesis is true.
Significance Level: The significance level $\alpha$ is the probability of making a Type I error in a hypothesis test.
Type II Error: A Type II error happens when a test fails to reject $H_0$ even though $H_0$ is false.
Effect Size: Effect size measures how large a difference or relationship is in practical, often standardized, terms.
Margin of Error: The margin of error $E$ is the maximum expected distance between a sample estimate and the true population value at a chosen confidence level.
Minimum Sample Size: Minimum sample size is the smallest $n$ needed to achieve a target margin of error or power under stated assumptions.

Common Mistakes to Avoid

Confusing $\alpha$ and $\beta$ is wrong because $\alpha$ measures false positives while $\beta$ measures false negatives.
Using a smaller sample size than the calculation suggests is wrong because it can lower power and make real effects harder to detect.
Forgetting to square the entire fraction in $n = \left(\frac{z_{\alpha/2}\sigma}{E}\right)^2$ is wrong because sample size depends on the square of both the critical value and the margin of error.
Using $p = 0.5$ as an estimate after better prior information is available can be inefficient because the best sample size calculation should use the most reasonable expected value of $p$ .
Treating statistical significance as practical importance is wrong because a tiny effect can be significant with a very large $n$ but still not matter in real life.

Practice Questions

1 A researcher wants to estimate a mean with $95\%$ confidence, known $\sigma = 12$ , and margin of error $E = 3$ . Using $z_{\alpha/2} = 1.96$ , find the required sample size $n$ .
2 A school survey estimates a proportion with $95\%$ confidence and margin of error $E = 0.04$ . If no prior estimate of $p$ is known, use $p = 0.5$ and $z_{\alpha/2} = 1.96$ to find $n$ .
3 A test has $\beta = 0.18$ . What is the power of the test?
4 Explain why increasing sample size usually increases power, even when the significance level $\alpha$ stays the same.

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