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manual:one_sample_proportion

# Statistical Power for One Sample Proportion

## Description

The power is to detect that a proportion is different from 0.5. The power calculation is based on the arcsine transformation of the proportion (see Cohen, 1988; p548). Specifically, for a given proportion $p$, the transformation is $\phi=2*arcsin(\sqrt{p})$. Given the null $p_0=0.5$, the effect size is defined by the difference after transformation $$h=2*arcsin(\sqrt{p})-2*arcsin(\sqrt{p_0})=2*arcsin(\sqrt{p})-2*arcsin(\sqrt{0.5}).$$ With the effect size, the power is calculated as $$\pi = 1-\Phi(C_\alpha - h\sqrt{n})$$ with $C_\alpha$ be the percentile from the standard normal distribution.

## Arguments

Among sample size, effect size, significance level, and power, one and only one can be left blank.

### Sample Size

Provide the number of participants. Multiple sample sizes can be provided in two ways. First, multiple sample sizes can be supplied separated by white spaces, e.g., 100 150 200 will calculate power for the three sample sizes. A sequence of sample sizes can be generated using the method s:e:i with s denoting the starting sample size, e as ending sample size, and i as the interval. For example, 100:150:10 will generate a sequence 100 110 120 130 140 150.

By default, the sample size is 100.

### Effect Size

The effect size to be used. Multiple effect sizes or a sequence of effect sizes can be supplied using the same method for sample size. By default, the value is 0.1.

The effect size is defined as $2*arcsin(\sqrt{p})-2*arcsin(\sqrt{.5})$. The effect size can be specifies as different from any constant in $2*arcsin(\sqrt{p})-2*arcsin(\sqrt{p_0})$.

The effect size can be calculated from proportion by clicking the “Show” button and then inputting the proportion directly. The proportion to be compared at H0 can also be specified.

### Significance Level (alpha)

The significance level (Type I error rate) for power calculation withe the default 0.05.

### Power

The power of the test.

### H1

Specifying the alternative hypothesis, can be “two.sided” (default), “greater” or “less”

### Power curve

Whether to generate the power curve.

### Note

A note (less than 200 characters) can be provided to provide basic information on the analysis.

Cohen, J. (1988). Statistical power analysis for the behavioral sciences (2nd ed.). Hillsdale,NJ: Lawrence Erlbaum.

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