Set your alpha level to establish how significant you want your results to be. To do this, consider how much funding you have available and how sure you are of the results of your study. Minimizing your sample size to the smallest it can be while keeping your results significant will save you money on extra participants and trials. Moreover, consider what the information will be used for. That is, if it is determining the relationship between rain and hot dog sales on the beach, you can require alpha to be slightly larger, like 0.20, or 20 percent. If your data is for the effects of a drug reducing symptoms of ADHD, you may want to keep your alpha to less than 0.05.
Find the test statistic relative to your alpha level. For instance, if you hypothesize that body temperature is greater than 98.3ºF, then you would find the test statistic for a normal alpha. However, if your hypothesis suggests an exact value, you have to consider the results possibly being greater than or less than 98.3ºF, meaning you would first divide your alpha in half. In the case of our hypothesis that the mean body temperature of humans is 98.3ºF, setting up an alpha of 10 percent, we would look up the Z-statistic from a normal distribution (Z) table that represents alpha/2 = 0.05. That Z-statistic is 1.645. Most statistics books provide a Z-table, but you can also use the one provided in the resources.
Calculate the margin of error, mean and standard deviation from a previous study. Before seriously considering to do research, statisticians first conduct a pilot study on a small sample, or they review literature on the subject. If a 90 percent confidence interval (alpha = 0.10) on body temperature was (98.0 to 98.6) with a mean of 98.3ºF and standard deviation of 0.733, then statisticians would build their hypothesis around a 98.3ºF temperature and use the previous information for sample size computation. The margin of error is the size of the confidence interval divided by two. In this case, the margin of error = (98.6 -- 98.0) / 2 = 0.3.
Determine the sample size needed, given you already have a margin of error corresponding to your alpha-level, a mean and a standard deviation from a previous study. The sample size, n, must be greater than [ (test statistic for the alpha-level) x (standard deviation) / (margin of error) ]². From our example, you get n > [ (1.645) x (0.733) / (0.3) ]² = (4.02)² = 16.2. So a sample size of 17 or more people should provide enough information to support an alpha level of 0.10.