Sample Size, $n$

Power, $1-\beta$

Type I error rate, $\alpha$

X-axis

min

max

Documentation
Test 1 Mean
Compare 2 Means
Compare k Means
Test 1 Proportion
Compare 2 Proportions
Compare Paired Proportions
Compare k Proportions
Test Time-To-Event Data
Test Odds Ratio
Test Relative Incidence in Self Controlled Case Series Studies
Other

## Calculate Sample Size Needed to Test 1 Proportion: 1-Sample, 2-Sided Equality

This calculator is useful for tests concerning whether a proportion, $p$, is equal to a reference value, $p_0$. The Null and Alternative hypotheses are

$H_0:p=p_0$
$H_1:p\neq p_0$

### Formulas

This calculator uses the following formulas to compute sample size and power, respectively: $$n=p(1-p)\left(\frac{z_{1-\alpha/2}+z_{1-\beta}}{p-p_0}\right)^2$$
$$1-\beta= \Phi\left(z-z_{1-\alpha/2}\right)+\Phi\left(-z-z_{1-\alpha/2}\right) \quad ,\quad z=\frac{p-p_0}{\sqrt{\frac{p(1-p)}{n}}}$$
where

### R Code

R code to implement these functions:

p=0.5
p0=0.3
alpha=0.05
beta=0.20
(n=p*(1-p)*((qnorm(1-alpha/2)+qnorm(1-beta))/(p-p0))^2)
ceiling(n) # 50
z=(p-p0)/sqrt(p*(1-p)/n)
(Power=pnorm(z-qnorm(1-alpha/2))+pnorm(-z-qnorm(1-alpha/2)))

### References

Chow S, Shao J, Wang H. 2008. Sample Size Calculations in Clinical Research. 2nd Ed. Chapman & Hall/CRC Biostatistics Series. page 85.