Forgive me if this seems like an elementary question, but no amount of googling has turned up a satisfying answer for me. From what I understand, a t-score should be used instead of a z-score when the sample size is considered small. I've also read that a t-score should be used when one is using the estimated variance, $s^2$, as opposed to the known population variance, $\sigma^2$. In addition, I've read that it is fine to use a t-score when $n$ is large and $\sigma^2$ is unknown, such as @gung's states here: Choosing between $z$-test and $t$-test. The reason being that the t-distribution closely approximates the normal distribution when the degrees of freedom are large, and therefore switching from a t-score to a z-score will likely make little difference. And then, finally, if $n$ is large, and $\sigma^2$ is known, it seems the consensus is to use a z-score, meaning you essentially end up with a flow chart like this:

enter image description here

(Note: this isn't my image, and I realize the "$n = 30$" rule is fairly arbitrary)

My question is, why would one use a z-score in the case where the sample size is large and $\sigma^2$ is known? If a t-score is to be used when $\sigma^2$ is known and $n$ is small, and the t-distribution closely approximates the z-distribution as $n$ becomes large, then wouldn't the choice between a z-score and a t-score make little difference? And if that is the case, why would anyone every use a z-score over a t-score? Is there some advantage to using a z-score I am missing, or perhaps a flaw in my understanding?


The image you used is wrong. If you know the population standard deviation then your statistic follows a normal distribution. When you do not know it, and you estimate from the sample than your statistic follows a t distribution and that is all. In your image and the text that follows the image you reverted yes with no.

Other than that when you know the exact distribution that there is no meaning of using an approximation. The fact that t distribution goes toward a normal distribution when the degrees of freedom is large is an approximation. Most probably this custom of using the normal approximation instead of t distribution when you have a large sample and estimate variance from the sample is an historical artifact. From the times when critical values were stored in printed tables, and computed values for t distribution for all degrees of freedom was hard if not impossible to find. But now we have computers.


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