Test preparation companies claim that their coaches can raise the SAT scores of high school students. A random sample of 89 students who took the SAT twice included 42 students who were not coached before their first SAT attempt but were coached before their second attempt. The sample also included 47 students who were uncoached for either attempt. Students who retake the SAT without coaching typically raise their scores, so we can test the claim that coaching has an effect on a student’s SAT score. The table below summarizes gains in SAT Math scores between the first and second attempt for each group of students (for example, the sample mean gain in SAT Math scores for the 42 students who were coached after their first attempt was 27 points, with a sample standard deviation of 11).

		Gain in SAT Math scores (in points)
	sample size (n)	sample mean	sample std. dev. (s)
Coached	42	27	11
Uncoached	47	23	10

Use the provided information to perform a two-sample t-test to determine whether the mean gain in SAT Math scores for students who were coached after their first attempt is higher than the mean gain in SAT Math scores for students who were not coached after their first attempt.

Based on the given information, using the appropriate formula and the provided Standard Normal Table (Z table), determine the p-value for this hypothesis test. You will need to calculate the test statistic first. Enter the p-value in the space below as a decimal rounded to four decimal places:

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Test preparation companies claim that their coaches can raise the SAT scores of high school students. A random sample of 89 students who took the SAT twice included 42 students who were not coached before their first SAT attempt but were coached before their second attempt. The sample also included 47 students who were uncoached for either attempt. Students who retake the SAT without coaching typically raise their scores, so we can test the claim that coaching has an effect on a student’s SAT score. The table below summarizes gains in SAT Math scores between the first and second attempt for each group of students (for example, the sample mean gain in SAT Math scores for the 42 students who were coached after their first attempt was 27 points, with a sample standard deviation of 11).

		Gain in SAT Math scores (in points)
	sample size (n)	sample mean	sample std. dev. (s)
Coached	42	27	11
Uncoached	47	23	10

Use the provided information to perform a two-sample t-test to determine whether the mean gain in SAT Math scores for students who were coached after their first attempt is not equal to the mean gain in SAT Math scores for students who were not coached after their first attempt.