# When to use a T-test instead of a Z-test (multiple regression)

So I’m going through quant, and I think I have a misconception about when to use the t-test vs. a z-test. The CFAI reading states “we use a t-test, rather than a z-test because we do not know the population variance of both b1 and b2…” I was under the impression that you can always use a z-test if the number of observations is above 30 OR (not and) the popualtion variance is known. I didnt know you needed both requirements. So obviously the CFAI books are right but can someone tell me why you need both conditions instead of just one? Thanks in advance!

You can always use a z-test, however the t-test is preferable. The population variance is needed in order to construct a t-test. But if you have a large sample, the sample variance can be a good proxy for the population variance and you can still go with the t-test. Hence, if the number of observations is above 30 or the population variance is known, you go with a t-test. If the number of observation is small and the variance is unknown, you’re stuck and you have to go with the z-test.

wait, it’s not the opposite of what you said? i thought if the population > 30 or pop variance is known you use a z-test, not a t-test. i don’t have my level 1 notes handy to confirm…

I confirm my previous message.

Looking at some notes from Schweser: “The t-distribution is appropriate when working with small samples (n<30) from populations with unknown variance and normal, or approximately normal, distributions. It may also be appropriate to use the t-distribution when the population variance is unknown and the sample size is large enough that the central limit theorem will assume the sampling distribution is approximately normal.”

olivier’s answer is all messed up. disregard. If you need to estimate the variance, you should be using a t-test. However, look at a table of z and t values. The critical values are pretty close for n > um, 30 or whatever looks pretty close to you. Using a t-test generally means that yu are assuming the observations come from a normal distribution. If you just blithely say that you are doing a z-test because n > 30 and disregard the sample variance issue, you are using the central limit theorem. That means you are doing an approximate test (i.e., your alpha, p-value, are approximate) and doing an approxmate t-test is just kinda weird. Your CLT approximation is made no worse by using the sample variance.

this is a level 1 discussion

lol. Sure. Sorry I offended you.

thanks joey & niblita. you both completely answered my questions. monki - i know that fundamentally, this is level 1 material, but i was confirming because this topic came up in level 2 CFA text. and indeed, i am studying for the level 2 test in june. don’t be such a snob. thanks!