To determine the minimum number of subjects needed for a z-test with a power of at least 80% and a type 1 error rate of 0.05, you can use a sample size calculator or perform the calculations manually.
Assuming that the population of differences is approximately normal, the z-test is appropriate to use. The z-test allows you to test whether the mean of a population differs significantly from a specified value (the null hypothesis) based on a sample taken from the population.
To perform the calculations manually, you will need to know the following:
- The desired power of the test (in this case, 80%)
- The type 1 error rate (also known as the alpha level or significance level) of the test (in this case, 0.05)
- The effect size, which is a measure of the size of the difference between the mean of the population and the null valueThe standard deviation of the population
Once you have this information, you can use the following formula to calculate the minimum sample size:
n = (Z_alpha/2 + Z_beta)^2 * (SD / effect size)^2
where:
- n is the minimum sample size
- Z_alpha/2 is the z-score corresponding to the type 1 error rate (in this case, 1.96)
- Z_beta is the z-score corresponding to the desired power of the test (in this case, 0.84)
- SD is the standard deviation of the population
- effect size is the size of the difference between the mean of the population and the null value, expressed in terms of the standard deviation
For example, let's say the desired power of the test is 80%, the type 1 error rate is 0.05, the effect size is 0.5, and the standard deviation of the population is 1. Using the formula above, we can calculate the minimum sample size as follows:n = (1.96 + 0.84)^2 * (1 / 0.5)^2
= 3.8236^2 * 2^2
= 14.52 * 4
= 58.08
Therefore, the minimum sample size needed for this test would be 59 subjects.
It's important to note that this is just a rough estimate, and the actual sample size may need to be adjusted based on the specific characteristics of the population and the research question being addressed.