What is the difference between a t-test and a z-test?

A z-test is a statistical test used to determine whether two population means are different when the variances are known and the sample size is large. The test statistic follows a standard normal distribution (a bell curve). In practice, you use a z-test if your sample size $n \ge 30$, or if by some rare chance you actually know the true standard deviation ($\sigma$) of the entire population.

A t-test is used when the population standard deviation is unknown and you must estimate it using your sample's standard deviation ($s$). Because estimating from a small sample introduces more uncertainty, the t-test relies on the Student's t-distribution. This distribution looks like a bell curve but has "fatter tails," meaning it requires stronger evidence to declare a significant difference. You use a t-test when your sample size $n < 30$ and $\sigma$ is unknown.

Sample size is the dividing line because of the Law of Large Numbers. As your sample size gets larger (specifically past 30), your sample standard deviation ($s$) becomes a highly accurate estimate of the population standard deviation ($\sigma$). As $n$ increases, the t-distribution mathematically morphs until it is virtually identical to the z-distribution. For small samples, though, that estimation is shaky, and the t-test corrects for that shakiness.

The most common mistake is looking at a word problem, seeing the words "standard deviation," and immediately assuming it's the population standard deviation ($\sigma$). Read carefully! If the problem says "a sample of 15 students had a mean of 80 and a standard deviation of 5," that 5 is the sample standard deviation ($s$). Because $n=15$ (which is less than 30) and you only have $s$, you must use a t-test.