# What is the difference between a test with assumption of standard normal distribution and t-test? And when to use each one?

I am studying statistics on my own and I am not understanding when to use a one-sample t-test or a test with assuming a standard normal distribution. As I understand, both are comparing a population mean with a sample mean, so, when should I use each one? And what is the difference between these two tests?

• There is no statistical test - named : test with a normal distribution. Normal distribution is an assumption for testing e.g. sample mean. Commented Feb 16, 2022 at 1:07
• use t-test when sample-size is "small" say , only 10 female soldiers in a large contingent of army. When sample is large e.g. blood corpscles in human blood samples , use standard normal test z-test to assess significance of mean estimate of red blood corpscles . Commented Nov 3, 2023 at 0:09

Do you mean when to compare to a t-distribution as opposed to a standard normal distribution? That is because, in the former case, you have to estimate the standard deviation. Chugging through the theory gives you that you compare to $$t$$, not standard normal.
The details of that should be covered in a statistics textbook. The intuition might not be. The way I think about it is that the standard deviation you estimate might be an underestimate. Thus, you use the thicker tails of the $$t$$-distribution to keep from getting a dishonesty small p-value.