The Professor's Response
I know that for a sample size to be valid it must be over
See the professor's answer below.
Hi Michael,
It looks like your message got cut off somehow. Please finish your thought below, and I will answer your question.
Thanks,
Professor McGuckian
Hi Michael,
Truthfully, the issue of choosing between Z and t is actually based on a formal rule which says that if the population standard deviation is known and you can assume normality, we are clerared to use the Z distribution. In many classrooms, they simplify this to say that as long as n > 30, we can assume normality because the Central Limit Theorem says that the sample mean will have approximately a normal distribution when n is larger than 30. In these settings, it is acceptable to use Z when n > 30.
If you knew the distribution of the underlying variable was normal and you knew the population standard deviation, you could use the z-curve even with a sample size of 20. The overall size of the population is less of a concern except for the concept of something called a finite-correction factor, which you shouldn't worry about for now.
If you are using the n > 30 rule to decide if you should use the Z-distribution which is technically incorrect (but suitable for classroom work), the sample size will need to be greater than 30 to use Z regardless of the size of the parent population and regardless of whether you know the population standard deviation.
Just a small clarrification though, a small sample size is not invalid. It just leads us to use the t-distribution instead of the Z distribution.