let me briefly anticipate that I am a total newby to the magical world of statistics. So, right now I'm preparing for a "quantitative methods for economics" test, but while doing my exercises I found myself wondering about what could be very stupid things.

here's the thing: I had two questions, which respectively looked like this:

QUESTION 1: A consultant hired by a University claims that the average student housing expense was 350€ per month. What are the null and alternative hypotheses to test whether this claim is accurate? b. The sample mean for student housing is 341.29€ and the sample standard deviation is 26.51€. Construct a 95% confidence interval for the population mean and evaluate the hypotheses of (a).

QUESTION 2: A poll by the German Sleep Foundation found that parents average about 7 hours of sleep per night. Researcher from a health insurance agency are interested that parents participating in their sleep program sleep longer than seven hours on average, and they would like to demonstrate this using a sample of parents. They conducted a simple random sample of n=110 parents. The conditions with respect to independence, sample size and skew are met. They find that the parents averaged 7.42 hours of sleep and the standard deviation of the amount of sleep for the students was 1.75 hours. a. Set up the null and the alternative hypotheses. b. Calculate the standard error of ̅. c. Compute the p-value and evaluate the hypotheses.

I have no problem with the calculations, because as you can see they are very easy in both cases. What is really not clear to me are these 3 things:

- In the first case, I construct the confidence interval (because it wasn't specified, I assumed a sample size of 40). Then I check the null and alternative hypothesis, respectively that: H0 = population mean is 350 per month, HA = population mean is not (any value different from) 350 per month. If I build the confidence interval at 95% and see that 350 is not included in it, I reject the null hypothesis. BUT If the value of 350 is indeed included in my interval, would I be able to reject HA? In the end, I still don't know what the true mean is, I only know that I can be 95% sure that it lies within that interval. So, in the latter case, should I just say that there is not enough evidence to reject HA and therefore we should accept H0? I think this might be the correct answer but I'm not sure.

- Building on this: the instructor of the course used the t-test for the second exercise. I don't get why in a case we are allowed to "evaluate the hypotheses" simply by constructing the confidence interval, while in the second we apparently need to use a t-test. I would like to know whether there is any general rule to use to understand when it's more appropriate to construct a confidence interval and when, on the other hand, you should use tools like the t-test. In the second case I would use a one-tail test, and my H0 = mean is not more than 7 hours sleep per night, while HA = mean is more than 7 hours sleep per night.

- How can you "compute the p-value"? I think this has something to do with the t-testing, but I'm not sure.

Thank you very very much for your help!