I took two statistics classes at college and I remember "hypothesis testing" to be a very powerful tool if only I could apply it correctly. It seems that while I was in college solving problems, I was always given the population's standard deviation to solve the question. However, what if I want to apply it in my job (i.e. track a particular process)? How can I obtain the true standard deviation of a process?
I am interested in normally distributed data. Let's say that I work at an industry that designs sprinkles, and I was interested to know if the product meets the specifications. The sprinkles were designed so that the average activation response time should be 25 (in a fire prevention system). Let's say I analyzed the times until 10 sprinkler samples activate in response to heat, and I got 27,41,22,27,23,35,30,33,24,27 (in seconds). Now I made the hypothesis test if the mean is equal to 25 or not equal to 25.
Best Answer
You are right - two assumptions of the classic z-test about a mean are hardly ever met in practice:
In not too small samples, these assumptions are not very important and the z-test is quite fine:
But in small samples (e.g. just ten observations as in your example), we cannot use these backdoors. Fortunately, a refinement of the z-test, the also very famous t-test takes care of issue 1: It correctly takes the additional uncertainty of the sample standard deviation (compared to the fixed $\sigma$) into account.
As a summary: In practice, whenever you can choose between z-test and t-test, always take the t-test. For large sample sizes, their results agree though and we could use the more simple z-test.
Final warnings: