11 mathematics competitors didn't perform too well on a practice exam. They were thus given extra time to prepare before the final competition. The differences between their scores in the final competition and the practice exam were the following: 10, 1, 6, 7, 5, 5,2, 3, 8, 9, 2

Assume that these 11 competitors were chosen randomly from a large population of competitors. It is known that score differences such as these among this population are normally distributed with mean and variance 33.65.

(a) Find an unbiased estimate for . (b) State suitable hypotheses to test the claim that the mean difference between final scores and practice scores is greater than zero. (c) Perform an appropriate hypothesis test given these hypotheses and calculate the P-value of the test. (d) Determine whether or not the above claim is supported at the 5% significance level. (e) What would a type I error and a type II error signify in this context?