True standard deviation (hypothesis testing)

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.

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Welcome @Faisal. I think we need some more info. What type of hypothesis are you interested in? What is the question you solved in college and how does it relate to the problem you're facing in your job? What is "track a particular process"? If you have the complete population of whatever you are interested in you can simply use the formula for a standard deviation. If not you can't get the true standard deviation, but you can estimate it. –  Momo Aug 30 '14 at 13:02
I am interested in a 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. –  Faisal Aug 30 '14 at 13:57
In order to apply the test, should I use the standard deviation of the sample? Because I know the in a z-test we can only use the standard deviation of the population which was always given in college problems. –  Faisal Aug 30 '14 at 13:59
@Faisal: In real life applications, the true standard deviation is never known and thus replaced by any consistent estimate. To account for this "extra" uncertainty, better use Student's t-test. It yields more reliable results for small to medium sample sizes. –  Michael Mayer Aug 30 '14 at 14:05
@Momo thank you! –  Faisal Aug 30 '14 at 14:10

You are right - two assumptions of the classic z-test about a mean are hardly ever met in practice:

1. The true standard deviation $\sigma$ is known.
2. The values come from a normal distribution.

In not too small samples, these assumptions are not very important and the z-test is quite fine:

1. We can replace the unknown $\sigma$ by its quite precise estimate.
2. Thanks to the almighty Central Limit Theorem, the test statistic of the test (standardized mean) is approximately normally distributed, even for quite "unnormal" observations.

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:

• Durations are usually not normally distributed but rather right skewed (even in your small sample there is such tendency).
• Not rejecting the null hypothesis does not mean that it holds.
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