Hypothesis test example

Among private universities in the United States, the mean ratio of students to professors is 35.2 (i.e., 35.2 students for each professor) with a standard deviation of 8.8. a. What is the probability that in a random sample of 25 private universities that the mean student-to-professor ratio exceeds 38?

Suppose a random sample of 25 universities is selected and the observed mean student-to- professor ratio is 38. Is there evidence that the reported mean ratio actually exceeds 35.2?

 For this we need P(X-bar > 38) so we get a Z score, where Z = (x-bar - mean)/(standard deviation/square root(n))
Z= (38-35.2)/(8.8/5)
Z = 3.2/1.76
Z = 1.82
Z(1.82) obtained from a standard normal distribution chart is .9656. Since we want P(X > 38) we take 1- .9656 = .0344
For part b
Ho: Mu = 35.2
Ha: Mu > 35.2
Test statistic : (x-bar - Mu)/(standard deviation/square root(n))
The population standard deviation is known, so use Z
Z = (38-35.2)/(8.8/5)
Z = 3.2/1.76 = 1.81
Decision rule: Critical value for .05 significance it 1.645, so we reject Ho if test statistic > 1.645 and do not reject otherwise
Decision is to reject Ho
Conclusion is that the student to professor ratio exceeds 35.2