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# Coaching company claims for raising SAT scores

Coaching compaines 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:
(all x's have bar over them)
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

Let's first ask if students who are coached increased their scores significantly.

(a) You could use the information on the Coached line to carry out either a two-sample t test comparing Try 1 with Try 2 for coached students or a matched pairs t test using Gain. Which is the correct test? Why?

(b) Carry out the proper test. What do you conclude?

(c) Give a 99% confidence interval for the mean gain of all students who are coached.