Step 1 :We are given that the SAT scores are distributed with a mean of 1,500 and a standard deviation of 300. We want to estimate the average SAT score of first year students at a college with a margin of error of 25 points for a 95% confidence interval.
Step 2 :We can use the formula for sample size in a confidence interval, which is \(n = \left(\frac{Z \cdot \sigma}{\text{Margin of Error}}\right)^2\). Here, \(Z\) is the z-score for a 95% confidence interval, which is approximately 1.96, \(\sigma\) is the standard deviation, which is 300, and the Margin of Error is 25.
Step 3 :Substituting the given values into the formula, we get \(n = \left(\frac{1.96 \cdot 300}{25}\right)^2\).
Step 4 :Calculating the above expression, we find that \(n = 554\).
Step 5 :Since we can't have a fraction of a student, we round up to the nearest whole number. So, we would need to sample 554 students to limit the margin of error of the 95% confidence interval to 25 points.
Step 6 :Final Answer: \(\boxed{554}\)