Math SAT scores (Y) are normally distributed with a mean of 500 and a standard deviation of 100. An evening school advertises that it can improve students' scores by roughly a third of a standard deviation, or 30 points, if they attend a course which runs over several weeks. The statistician for a consumer protection agency suspects that the courses are not effective. She views the situation as follows: H0: μ_0 = 500 vs. H1:μ_a = 530.
a) The statistician send 49 students for the school and after graduating from the course, the 49 participants take the SAT test and score an average of 520. Is this convincing evidence that the school has fallen short of its claim? What is the p-value for such a score under the null hypothesis?
b) What would be the critical value under the null hypothesis if the size of your test were 5%?