What we really want to know is whether coached students improve more than uncoached students, on average, and whether any advantage is large enough to be worth paying for. Use the information above to answer these questions:

a. How much more do coached students gain, on average, compared to uncoached students? Construct and interpret a 99% confidence interval.

b. Does the interval in part (a) give convincing evidence that coached students gain more, on average, than uncoached students? Explain your answer.
c. Based on your work, what is your opinion: Do you think coaching
courses are worth paying for?

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 the first and second tries, we have these summary statistics:

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Try 1 Try 2 Gain 1. SX 305 =0.200 1526 SX SX 305 305 0.200 1526 305 = 0.200 1526 305 305 0.200- 1526 0.200 0.200 : 1526 1526 Coached 427 500 92 529 97 29 59 Uncoache 2733 506 101 527 101 21 52