Hypothesis Testing Help

[kolorlover]

I have some hypothesis testing questions that I need guidance on. Question #1, feels like I did it right. It would be great if someone can confirm it? Question #2, stuck at one part. Help need please?

Question #1:

An oceanographer wants to test, on the basis of the mean of a random sample of size n = 35 and at the 0.05 level of significance, whether the average depth of the ocean in a certain area is 72.4 fathoms, as has been recorded. What will be decided if the sample mean is 73.2 fathoms? One can assume from information gathered in similar studies that σ = 2.1 fathoms.

Here's what I did:

In this question, we are going to use Z-score since the sample size is greater than 30 and classical hypothesis testing will be used.

Z = Xbar – µ / σXbar = Xbar – µ / σ / √n

Z = 73.2 – 72.4 / 2.1 / √35

Z = 0.8 / 0.354964787

Z = 2.253744679

Z = 2.3

For significance level of 0.05, Z is greater than or equal to 1.64, hence the null hypothesis should be rejected. The value is 2.3 fathoms, which is greater than the population mean of 2.1 fathoms.


Question #2:

A researcher in the social sciences is concerned that a sample of respondents to a mail-out survey may not be representative of the target population under study. Specifically, the researcher is concerned that the sample of respondents are significantly younger than the population average of 37. The social scientist obtained a mean age of 22.4 years, and a standard deviation of 6.8 for a sample of 40 respondents. What can the social scientist conclude if the probability of a Type I error is to be at most 0.01?

This is what I'm doing:

Z = Xbar – µ / σXbar = Xbar – µ / σ / √n

Z = 22.4 – 37 / 6.8 / √40

Z = -14.6 / 1.075174404

Z = -13.57919231

Z = -13.6

And I don't know what to do next. Like, do you need to find the probability from the Z-score table? But then isn't the value of -13.6 too big?

[sanddevil]

So, first off:

Question #1: this is a minor point, but you're only reporting a one-sided p-value, which assumes that you already know that the ocean depth can ONLY be greater than 72.4 fathoms. You should instead use 1.96--Z(1-alpha/2) instead of Z(1-alpha)--as your cut-off for rejection of the null. In this particular case, it doesn't change the result of your overall test, but in general, you should ALWAYS report two-sided values, because they are more conservative.

Question #2: Not much has changed here except for your alpha. 1.645 was Z(1-alpha) in your last problem, with the assumption that alpha = 0.05, and you compared 2.3 to 1.645. In this case, your prof is giving you a new alpha (0.01), so you need to go to your standard normal table, figure out which Z value cuts off the graph at 0.01, and then compare -13.6 to this value.
In this case, your prof has directed you to do a one-sided comparison with the words "...are significantly YOUNGER than the population average..." which means you can get away with comparing -13.6 to Z(1-alpha) instead of Z(1-alpha/2) as in question #1.

Hope this makes helps!