The z-test is used when the sample size is greater than 30 and the standard deviation is known; the t-test is used when the sample size is less than 30 and the standard deviation is not known (Math Meeting, 2013). When the standard deviation is known we use that to calculate the probability.
The t-test should be used when a sample is normally distributed and is used to compare the means of two different groups (Grove & Cipher, 2017).
If we have a sample size of 75 and we know the standard deviation, we would use the z-test.
If we have a sample size of 12 and we do not know the standard deviation, we would use the t-test.
If we have a sample size of 20 and we know the standard deviation, we would use the z-test.