Alright so I have some confusion on when to use specific tests and the z vs t test.

Given this example (not my homework) could someone please clarify.

Alright say you have a random sample of size 200. You find the sample mean to be 10 and the sample standard deviation to be 5.

What would be the answer to these questions?

1. What do you think the population sd and mean are?

Construct a 95% confidence interval for mu given sigma is know

Construct a 95% confidence interval for mu given sigma is unknown using s(sample sd)

First major question, what is the difference between 2 and 3? My reasoning is that for 2 you use the z alpha/2 value and for 3 you use the t value, is this correct?

Assume you would like to have your estimate to be with 0.5 units from µ with a 95% CI, what should the sample size be?

For this question should I use the z value or t value? I know how to calculate after that.

Ignore the hypothesis values of 3 and 2, again I am unsure whether to use a t or z value here.

5.Please construct a test statistic to test H0 : µ ≤ 3 versus Ha : µ > 3, and provide the formula for the test statistic. At a probability of Type I error α = .05, do you reject H0? Why or why not?

Consider H0 : µ = 2 versus Ha : µ != 2. At a probability of Type I error α = .05, do you reject H0?Why or why not?

Please construct a test statistic to test H0 : µ ≤ 3 versus Ha : µ > 3, and provide the formula for the test statistic. At a probability of Type I error α = .05, do you reject H0? Why or why not? Consider H0 : µ = 2 versus Ha : µ != 2. At a probability of Type I error α = .05, do you reject H0? Why or why not?

This is bonus but I would be happy to hear if you know something about it

For testing H0 : µ = 2 versus Ha : µ != 2, if s is smaller than σ and the test statistic is based on s instead of on σ, is it easier to reject H0? Why or why not?

Overall I am confused on when to use z vs t and the conceptual notions behind it. Any help is appreciated.

Thank you for your time