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QUESTION

# If you don't know the population standard deviation, can you perform a one-sample z test for the mean?

No. You should use student's t-test.

I'm assuming the population has a normal distribution. The statistic used in the z-test rely on the population standard deviation, so you should use student's t-test, wich depends on the sample standard deviation instead.

The statistic for student's t-test is \frac{\barX - \mu_0}{S/\sqrtn , wich has a t distribution with n-1 degrees of freedon.

Also note that since the t-distribution converges to a normal distribution when n goes to infinite, if you have a large sample, you can perform t-test similar to the z-test, as the test statistic will have normal distribution.