Fundamentals of Statistics 3: Sampling :: The 1-Sample Z-test: You'll probably never use it
The 1-sample z-test allows us to compare a sample mean to a population mean to determine how likely it is to obtain that particular mean. To use the test we need to know the standard deviation of the population. In practice, you pretty much never know the population standard deviation. If you did, then you probably wouldn't be sampling in the first place!  Examples of when we would likely know the population standard deviation are for things like IQ scores, SAT or ACT scores (those standardized college entrance exams) and heights and weights.

We can use the 1-sample z-test on any measure that has been collected for the entire population of interest, provided that we know the standard deviation. That's why I've been using heights and IQ's for many of the examples (in case you were wondering). Photo By: po go re lova You're unlikely to use the 1-sample Z-test outside of a statistics class since you almost never know the population SD.
So what's the point in learning it if you almost never use it (you might begrudgingly ask yourself)? The reason is that in math and in statistics it is easy to get lost in all the computations. To help you understand the bigger picture, statistic teachers across the world think it is easier for new students to understand the general concept of using the normal distribution to understand probabilities of events if you don't have to get bogged down by working out the additional complexities of what you do if you don't know anything about the population.  I generally agree with this idea too and we see it a lot in science. For example, remember in middle-school science class how you were taught to think of the atom like a mini-solar system?  The analogy was helpful for us to get an idea about how the electron spins around the nucleus of an atom. In real life it turns out that this model is flawed and there is no mini-solar system (Heisenberg etc).

So when I first learned statistics I wish someone had told me that you'll probably never use the 1-sample z-test and if you find yourself using or considering it, then you're probably in the wrong place--go to the t-test instead (unless of course you're in a stats class then you'll use it quite a bit). As we'll see, you'll spend most of your time using the 1-sample t-test, which is the appropriate test when you don't know your population standard deviation.

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