Researcher at a major clinic wishes to estimate the proportion of the adult population of the US that has sleep deprivation. How large a sample is needed in order to be 98% confident that the sample proportion will not differ from the true proportion by more than 5%?

If you know the standard deviation σ of the population, and you want to estimate the mean μ to within a given margin of error E in a 1−α confidence interval, here’s how to find the required sample size n:

transforms to

example: You want to estimate the average hourly output of a machine to within ±1.5, with 90% confidence. Based on historical data, you have reason to believe that the standard deviation of the machine’s hourly output is 6.2. How large a sample do you need?

Solution:Note first that this is not a realistic situation. it's pretty unlikely that you would know the standard deviation of a population but not know the mean of that population. However, statistics texts always begin with this case because it’s the simplest way to demonstrate the principles. You leave Perfectland and enter Realityville in the other cases