Coaching companies claim that their courses can raise the SAT scores of high school students. Of course, students who retake the SAT without paying for coaching generally raise their scores. A random sample of students who took the SAT twice found 427 who were coached and 2733 who were uncoached. Starting with their verbal scores on he first and second tries, we have these summary statistics: Try 1 Try 2 Gain
n x¯¯¯ s x¯¯¯ s x¯¯¯ s
Coached 427 500 92 529 97 29 59
Uncoached 2733 506 101 527 101 21 52

Construct, using the conservative estimate for degrees of freedom, a 98% confidence interval for the difference, μc−μu between the mean gain of all students who are coached and the mean gain of all students who are uncoached.