Whether to use z test or t test

#1
hi team,

I got the following question given below. According to me this should be solved using z test as the sample size is more than 30. But they are using t test. What is the correct method z test or t test? Please advice

A social worker wants to know whether the mean annual salary of her clients
matches the mean annual salary for all city residents, which is $28,000, or whether
it is less. She obtains a random sample of her clients salaries of size 36. The
mean of the sample is $27,500 and the standard deviation is $1,200. Perform a
hypothesis test at the 5% level of significance

Thanks
Simi
 

hlsmith

Omega Contributor
#2
Funny enough, over the years I have wondered what the general rule was as well. I imagine it is more of a discipline thing. Though, I think there may be a pseudo-sample size rule. The t-distribution has slightly longer tails if I remember right, but with increasing sample size will converge toward the standard normal.


I would be interested to see if anyone else has some input.
 

Dason

Ambassador to the humans
#4
I feel like these rules are just holdovers from the old days when it was much more difficult to do something like a t-test than it is now. Can you do a z-test instead of a t-test when your sample size is large enough? It doesn't really make a difference so some people go and do that. It's just seems silly to me though since we actually can do the t-test and there is no real reason to opt for z instead of t in a situation like this.

With all that said most likely the assumptions for either test aren't fully 100% met so they're both just approximations to the truth anyways.
 

hlsmith

Omega Contributor
#5
Dason, I totally agree with you. I also thought this may be a hangover from previous approaches. I just called it a discipline thing because I know someone not in an analytic field that runs a single test on occasion looking for non-representativeness of a sample and they use a Ztest. I feel like the supreme court or some governing body said in their field, hey this is how you can do it.


Though, I have always wondered why. That is the reason I am intrigue by this question. Whether there is a sample size (general) rule or theory behind it. Or is one test more forgiving to normality assumptions, etc..
 
#6
Thankyou all for the reply. I am new to statistics. I read in one book which says, if population standard deviation is unknown and the sample size is less than 30, then we go for t test and in other case we go for z test. But when i was trying to solve some problems which i saw online, i saw this is not the case. So i was looking for some guidance.