Statistic question help?

Question

I'm struggling to understand when do we use the t or z test.Must they always specify that it is the population variance/standard deviation in order for me to choose the z test?If they only tell me that it is the variance or standard deviation only(not saying whether it is of a sample or of a population) i must use t-test?

Answer 1

Both tests can only be used when the underlying distribution is Normal.

Z tests are used when you KNOW for sure the population variance/standard deviation. It will be given to you.

T tests are used when you must calculate for yourself the sample variance/standard deviation.

If they use "s" to denote the var/s.d., then it's sample. If they use "sigma" to denote the var/s.d., then it's population. Depending on the question, if they don't tell you it's a sample var/s.d., then you should be able to assume that it's population. Just make sure to write down this assumption.

Answer 2

you may use the z test in the following cases:

1) your data is from a normal population and the population variance is known.

2) your data is from a large enough sample that the central limit theorem holds and we can assume the mean follows the normal distribution.  (note that the z is okay to use, but the t would be better here.)

you use the t test if:

1) your data is from a normal population and the population variance is unknown, so you use the sample variance.

2) your data is from a large enough population the central limit theorem holds and you are using the sample variance. (z would be okay, but t is better in this case.)

if you are not told that the given variance/std dev is sample or population then assume it is sample.  It's a safer move because in real life we will not know the population variance.