Suppose that in the absence of special preparation SAT mathematics scores vary normally with a mean of 475 and a population standard deviation of 100. One hundred students go through a rigorous training program designed to raise their SAT mathematics scores by improving their mathematics skills. The students' average score after the training program is 510.9. Test the claim that the training program improves students' average SAT mathematics scores. Test at the 5% level of significance.

a) State the null (H0) and alternate (Ha) hypotheses (indicate the claim).

b) If the null hypothesis is true, the sampling distribution of the population mean has an approximately Normal distribution with Mean _______________ and Standard Deviation ____________________.

c) Calculate the test statistic (standard score).

d) Calculate the P-value.

e) Make a decision to reject or fail to reject the null hypothesis. Summarize the final conclusion in the context of the original claim.