Question 5
๒0/1pt り5 (i) Details

SAT scores are distributed with a mean of 1,500 and a standard deviation of 300 . You are interested in estimating the average SAT score of first year students at your college. If you would like to limit the margin of error of your $95 \%$ confidence interval to 25 points, how many students should you sample?

Make sure to give a whole number answer.

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Step 1 ：The problem is asking for the sample size needed to estimate the average SAT score of first year students at a college with a margin of error of 25 points and a 95% confidence level.

Step 2 ：The formula for the margin of error in a confidence interval is: \(E = Z * (\sigma/\sqrt{n})\), where \(E\) is the margin of error, \(Z\) is the Z-score (which corresponds to the desired confidence level), \(\sigma\) is the standard deviation, and \(n\) is the sample size.

Step 3 ：We can rearrange this formula to solve for \(n\): \(n = (Z * \sigma / E)^2\)

Step 4 ：We know that the standard deviation (\(\sigma\)) is 300, the margin of error (\(E\)) is 25, and the Z-score for a 95% confidence level is approximately 1.96.

Step 5 ：We can substitute these values into the formula to find the required sample size: \(n = (1.96 * 300 / 25)^2\)

Step 6 ：Calculating the above expression, we find that \(n = 554\)

Step 7 ：Final Answer: The required sample size to limit the margin of error of the 95% confidence interval to 25 points is \(\boxed{554}\).