1. Researchers want to test whether an Indian child's school grades are influenced by his or her caste. They believe graders simply take 10 points off the score of an exam written by a student belonging to a lower caste. In order to test this hypothesis, they obtain the scores of children of higher and lower castes, on a test on which children are generally believed to do equally well. In their sample, among the 45 children belonging to the higher castes, the average grade was 74, with a standard deviation of 12. Among the 41 children belonging to the lower castes, the average grade was 61, with a standard deviation of 11.

(a) (8 points) Test the hypothesis that the difference in the mean scores of children of higher castes and the children of lower castes is 10 points, against the two-tailed alternative hypothesis at 1% significance level. Use a t-test with equal standard deviation.

(b) (4 points) Using the pooled standard deviation calculated before, calculate a 90% confidence interval for the difference in mean score of children of lower castes and the mean score of children of higher castes.