Is a one sample t test the right approach for this question? This is an example from my book that I'm trying to figure out but I'm totally lost. Can someone complete this step by step so I understand?

A new meditation technique based on making barnyard animal noises promises to calm people, but you might be skeptical. You can imagine that it could cause stress to sit around oinking and mooing. To test the therapy, you ask 16 people to make animal noises, then ask them to rate emotion on a scale from 1-7, with 1 representing “Very Calm,” and 7 representing “Very Stressed.” You happen to know that people who don’t oink and moo have an average emotion rating of 4, with a standard deviation of 2.57. Below are the ratings from the 16 people who made animal noises.

 4  5   6   5   4   3   7   4   

 3  4   4   3   5   6   5   4


Write the null and alternative hypotheses for this example.
  Write the specific type of analysis you will use.
  Zobt? Z crit?
  Provide a statement of a significant or nonsignificant effect.
  Write the obtained value.

I believe my null and alternate hypotheses are: 
$\text{H}_0: \overline{X}= μ\quad\text{vs}\quad    \text{H}_1:\overline{X} \neq μ$ 
for $t$ I did  $\frac{4.5-4}{2.57/\sqrt{16}}$ and got $.778$? Is this correct?
My $z_\text{crit}$ would be $2.131$?
 A: Let's begin with the requisite hints and guidance.
A number of things you have there are wrong. Let's look at a few:


*

*You're given several pointers to this not being a t-test (the fact that you're given a known standard deviation and the repeated reference to $z$).

*Hypotheses don't involve sample quantities (like $\overline{x}$), but population quantities. It's about a comparison of the (unknown) population mean rating of people making animal noises (for which you have a sample) against the known population mean rating of people not making animal noises. 

*your critical value should not be from t-tables

*you haven't stated a significance level. Since none is given in the question, this should be one of the early things to do, and certainly before you calculate a test statistic.

*you should explicitly state the assumption you made about $\sigma$ for the animal noise group (that it's the same as the population s.d for the no-noise group). If you don't make that assumption (and it isn't going to be easy to justify - the question offers no justification, as Nick points out below), you would use the one-sample t-test, but in that case your calculation of the test statistic would change)
One the plus side:


*

*you're right that it's one-sample. 

*I believe your calculation of the test statistic is correct for the z-test; you found all the necessary pieces of information. However the fact that you didn't compute a sample standard deviation while doing so should have been a big clue that it wasn't a t-test.
A: Your hypotheses look good to me, but, "You can imagine that it could cause stress to sit around oinking and mooing," sounds like a directional hypothesis, so a one-sided hypothesis is also worth considering. Then again, one-sided hypothesis tests seem to be rather controversial, so maybe not!
Your calculation of t is incorrect. You're right that $\bar X=4.5, \mu=4$ and $N=16$, but the population standard deviation in the question is sort of a red herring. You need to use the sample standard deviation for a one-sample t-test.
I don't understand why a $Z_{\rm critical}$ is requested. A Z-test is inappropriate without knowledge of the experimental population's standard deviation. It doesn't seem justified to assume the given SD parameter for the control population would apply equally to the test population. For one thing, without controlling for personality differences like extraversion or self-monitoring, making animal noises might increase variability in stress levels by allowing some people to release stress by acting silly while making others more stressed due to embarrassment. By the way, the question implies that the ordinal data from the Likert scale for stress assessment can be treated as continuous, which is also fishy...
Furthermore, even if a $t_{\rm critical}$ was requested instead, the appropriate value to provide would depend on the choice of false positive error rate $\alpha$ for the test. It's conventional to choose $\alpha=.05$, so if you haven't been given any other reason to choose a different error rate, you probably won't go wrong (for grading purposes at least) with $t_{(\alpha=.95,df=15)}=2.131$ as long as you don't call it a $Z_{\rm critical}$.
You don't seem to have attempted the last part, "Provide a statement of a significant or nonsignificant effect." Give it a shot. It shouldn't be too hard as long as you choose your words carefully (don't get too creative, and be careful not to claim more than your results actually tell you) and understand which conclusion is appropriate when your test statistic is higher vs. lower than the critical value.
