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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 (x) of a plant  in some defined period by \delta.  The researcher assumes:

  • x 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 \alpha
  • the desired power (1 -type II error) is 1 -\beta
The null hypothesis (H_0) is that the mean of the control (0) and treated  population(1) are equal:
H_0: \mu_0=\mu_1
The alternate hypothesis is:
H_1: \mu_1-\mu_0=\delta
Consider the standardised normal distributions:
P(Z>Z_{\alpha}|H_0)=\alpha
P(Z>Z_{1-\beta}=-Z_{\beta}|H_1)=1-\beta
Under the assumption of equal variances (\sigma^2), the variance of the difference is 2\sigma^2/n.
The critical value on the raw scale must coincide. Back transforming the standardised variable  yields:
x_{crit} =0 +Z_\alpha \sigma\sqrt{2/n} = \delta -Z_\beta\sigma\sqrt{2/n}
Solving for n:
n =\frac{2(Z_\alpha +Z_\beta)^2\sigma^2}{\delta^2}
 I have used a one tailed type I error, for two-tailed just substitute \alpha/2 for \alpha from diagram to text.
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Categories: Mathematica, Mathematics
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