Question

I've got a couple questions pertaining to scenarios involving this question:

Construct 90% confidence intervals for the true average wait time for customers at both the Hialeah and Miami Gardens offices. Analyze the results and indicate which office is better in

terms of average customer wait time.

The resulting intervals are found with a t score

Hialeah: (20.947, 25.053)

Miami: (17.092, 22.908)

The professor said: "The CI overlaps so it cannot be said that one office better than the other"

My question is if the CI's didn't overlap how would I determine which one is better than the other?

Also a T-score was used to evaluate these intervals since the given population was 25. Are there any scenarios with a small population (n) < 40 where a Z score would be used?

Answer 1

This is about wait time. So, if some office has less waiting time then that's better one compare to other. So, if we have CI like:

Hialeah: (15, 17)

Miami: (18, 19)

Now we see that there is no overlap and upper limit of Hialeah is 17 whereas lower limit of Miami is 18. So, we can say that customers have less wait time in Hialeah office so that one is better. I hope this point is clear.

Now, second question when can we use z-score even if sample size is small. Yes, if it's given that the sample is taken from population which follow normal distribution and it's population standard deviation is given then we use z-score irrespective of sample size. Main point is 'population standard deviation is known or not' if known use z-score if not then use t-score for small sample size.

I hope this will help. Thank you.