Score: $4.9 / 10$
$6 / 10$ answered
Question 9

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?

Question Help: $\triangle$ Message instructor
Calculator
Submit Question

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: \(\boxed{554}\)