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High school students who take the SAT Mathematics exam second time generally score higher than upon their first try. The change in score has Normal distribution with standard deviation ? = 50. A random sample of 1000 students gain average of = 22 points on their second try.

a. Compute the 90%, 95% and 99% confidence intervals for ?(Mu).

b. Compare such three intervals. How does rising the confidence level affect the width of the confidence interval?

c. What are the margins of error for samples of size 500, 1000, and 2000 using a 99% confidence level? How does increasing sample size affect the margin of error of a confidence interval? Show your work and justify your answers.

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