## How many?

Sample size estimation for various research objectives is a complex exercise. In this post, I look at the usual derivation for “back of the envelope” calculations of sample size.

Consider a researcher who believes that a chemical added to a solution will increase the height () of a plant in some defined period by . The researcher assumes:

- follows a normal distribution (see figure)
- the variance of the control and treated distributions are the same (see figure)
- the desired type I error is
- the desired power (1 -type II error) is

The null hypothesis () is that the mean of the control (0) and treated population(1) are equal:

The alternate hypothesis is:

Consider the standardised normal distributions:

Under the assumption of equal variances (), the variance of the difference is .

The critical value on the raw scale must coincide. Back transforming the standardised variable yields:

Solving for :

I have used a one tailed type I error, for two-tailed just substitute for from diagram to text.

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Categories: Mathematica, Mathematics

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