Sample Size Calculation - Means

Suppose an experiment is to be conducted where a treatment of some sort is to be applied to a population. The investigator is interested in knowing the minimum sample size that should be randomly selected from the population to detect a change in the mean of the population as a result of the treatment. The statistical factors that influence the sample size are:
- the value selected for , the Type I risk (the risk of rejecting a true null hypothesis, )
- the value selected for , the Type II risk (the risk of accepting the null hypothesis, , as true when, in reality, it is false)
- the value of the population standard deviation, σ (this is often an estimate, based on judgement and experience)
- the size (relative to σ) of the effect to be detected by the sample

The Type I risk is that of getting a false positive from the sample. In other words, there is a probability of that the sample will indicate that there is a difference of at least when, in fact, there is not.
The Type II risk is that of getting a false negative from the sample. In other words, there is a probability of that the sample will indicate there is not a difference of at least when, in fact, there is such a difference. The ability to detect a difference when there actually is one is called the power of the test and is equal to .

Additionally, whether the test of the hypothesis is a one-sided or two-sided test has an effect on the sample size. A one-sided test would, for example, look at whether there is an increase OR a decrease in the mean (not both), whereas a two-sided test would consider whether the mean is EITHER more than or less than the value in the null hypothesis.

When using this Demonstration, you should mentally formulate a null hypothesis and an alternative hypothesis (either one- or two-sided). Then the appropriate control selections can be made and the resulting sample sizes can be examined.
Typical values for and are 0.05 and 0.20, respectively, resulting in a confidence of 95% and a test power of 80%.
Note that looking for a small difference, , drives a large sample size, as would be expected.