solution

There is much controversy about whether coaching programs improve scores on SAT exams by more than a minimal amount. Although this exercise wonâ€™t settle the argument, it does present a practical use of a test of significance. The College Board reports that coaching programs improve SAT mathematics scores by about 25 points on average. A sample of 50 students in a â€œcontrolâ€ group who took the math portion of the SAT as juniors and then again as seniors had an average gain of 13 points (reported to be about the national average gain without coaching), with standard deviation of about 30 points. A sample of 9 students who were coached between their two exams had an average gain of 60 points, with a standard deviation of 42. [Source: Jack Kaplan, â€œAn SAT Coaching Program That Works,â€ Chance 15, no. 1 (2002): 12â€“17.]

a. Is their evidence that the difference in mean point gain between the coached group and the control group exceeds the 25 points that is to be expected (according to the College Board)?

Â b. Another sample of 12 students from a coaching program had an average gain of 73 points, with a standard deviation of 42. Is there evidence that the difference in mean point gain here exceeds the 25 points that is to be expected?