1. We assume that heights are independent and follow a normal distribution with unknown mean and SD. In this case, (Xbar - mu)/(s/sqrt(n)) is a standard normal random variable, and we can use it to build a confidence interval:

Xbar +- t* s/sqrt(n)
We use the t* because sigma is unknown and estimated by s.  t* follows a t distribution with n-1 or 37 df.  The critical value for a 95% confidence interval is 2.03.  Because n is moderately large, it is forgiveable to use 1.96 -- the normal approximation-- although as you can see you'll get a CI that is too small.  (I'm not sure what values I gave you for the quiz. If you had only the table in the back of the book, which only handles df <= 30, then the normal approximation is the best you could do.  But if you had better tables or a computer, then you should use the 2.03 value.)

67.868 +- 2.03*2.8891/sqrt(38)

67.868 +- 0.951375

(66.9166, 68.8194) is a 95% confidence interval for mu.
 
 

2. No, this is not a good estimate, because our sample is not a representative sample. To get a representative sample, we would need to take a random sample of all college students.