Questions: 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?

Solution Steps

We are given the following parameters for the SAT scores:

• Population mean (\( \mu \)) = 1500
• Population standard deviation (\( \sigma \)) = 300
• Desired margin of error (\( E \)) = 25
• Confidence level = 95%, which corresponds to a Z-score (\( Z \)) of approximately 1.96.

The margin of error for a confidence interval for a population mean is given by the formula:

\[ E = Z \times \frac{\sigma}{\sqrt{n}} \]

Where:

• \( E \) is the margin of error,
• \( Z \) is the Z-score,
• \( \sigma \) is the population standard deviation,
• \( n \) is the sample size.

To find the required sample size (\( n \)), we can rearrange the formula:

\[ n = \left( \frac{Z \times \sigma}{E} \right)^2 \]

Substituting the known values into the rearranged formula:

\[ n = \left( \frac{1.96 \times 300}{25} \right)^2 \]

Calculating the expression step-by-step:

1. Calculate \( Z \times \sigma \): \[ 1.96 \times 300 = 588 \]

2. Divide by the margin of error: \[ \frac{588}{25} = 23.52 \]

3. Square the result: \[ n = (23.52)^2 = 552.5504 \]

4. Round up to the nearest whole number: \[ n = 553 \]

Final Answer