How to use the Z-test or T-Test given the sample of a non-normal population

#1
Hi Everyone, I've tried my best to stay away from forums because I like to try to learn things on my own, but this is the third assignment I've had in this class and I'm baffled by it.

I don't want to copy it verbatim in case the prof is watching. We are given the sample data of non-normal population. The prof is asking us to determine whether it's likely that the population mean is 38. Also, he has stated that we should choose either the Z-test, t-test, or f-test. He also said it's not likely the Z because z is only for normal distributions.

I immediately worked out the descriptive statistics:

Xbar = 38.39
Sample standard deviation = 0.42
Count = 31

And starting working out the critical values for the T-Test, but I realized that I can't do this because if the population is non-normal, therefore the sample should also be non-normal, correct?

We learned about the central limit theorem in class, but i'm not sure how it would apply here. Perhaps I could use the mean of this sample to create a distribution of means? However this doesn't help me because the count would only be 1, and therefore the degrees of freedom would be 0. I couldn't use the Z-test here either because I don't know the population standard deviation.

I hope someone can help me. The raw data is below.

Thanks!

32
35
35
36
36
36
37
37
37
37
38
38
38
38
38
39
39
39
39
39
39
39
40
40
40
40
41
41
41
43
43
 
#2
Anyone?

I may just use the t-test directly and hope for the best. It seems the sample data is normally distributed so this might be my saving grace.
 

rogojel

TS Contributor
#3
Hi,
two quick observations:

1. the Z test is not applicable because you do not have an independent estimate of the population standard deviation. If you estimate the std dev from the sample you need a t distribution, which will be fairly close tonthe normal for samples above roughly 30, BTW.
2. If you worry about the population being non-normal you should go for a non-parametric test. As your choice is t or Z, with f an obvious trap question for the unwary, you should go with the t-test as it is quite robust against deviations from normality and the Z is unfeasible anyway.

regards
 
#4
Hi,
two quick observations:

1. the Z test is not applicable because you do not have an independent estimate of the population standard deviation. If you estimate the std dev from the sample you need a t distribution, which will be fairly close tonthe normal for samples above roughly 30, BTW.
2. If you worry about the population being non-normal you should go for a non-parametric test. As your choice is t or Z, with f an obvious trap question for the unwary, you should go with the t-test as it is quite robust against deviations from normality and the Z is unfeasible anyway.

regards
Thank you! I ended up using the t-test like you said!!