Suppose we want to assess the effectiveness of a short course on nutrition. Eight students were given a test before and after taking a short course on nutrition. The data are shown below. Student 1 2 3 4 5 6 7 8 Before After Improvement 50 56 6 71 78 7 87 100 13 75 71 -4 82 95 96 91 63 80 13 5 16 79 84 4 a) Why the above sample observations are treated as matched pair (dependent) samples? What is the difference between a matched pair (paired) scenario and a two-sample scenario? b) Let the ith student's before (B) and after (A) score be denoted as zB) and r). Also, let d; denotes the ith student's score difference, (23) I Suppose we are interested in an interval estimation for population mean difference 5. As- suming the differences are normally distributed, please construct a 95% confidence interval for us. You may use ZTest() or t.test() to verify your answer, but replicate the values mathematically yourself and show your work as well.