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The question asks how many students the administrator should sample to limit the margin of error of a 98% confidence interval for the average SAT score to 20 points. Here's how we can solve it:

### Given Data:
- Standard deviation (\( \sigma \)): 300
- Desired margin of error (\( E \)): 20
- Confidence level: 98%

### Formula:
The formula to determine the required sample size (\( n \)) for estimating a population mean with a given margin of error is:

\[
n = \left( \frac{Z_{\alpha/2} \cdot \sigma}{E} \right)^2
\]

Where:
- \( Z_{\alpha/2} \) is the critical value from the standard normal distribution for a 98% confidence level.
- \( \sigma \) is the population standard deviation.
- \( E \) is the margin of error.

### Step 1: Find the critical value \( Z_{\alpha/2} \) for a 98% confidence interval.
For a 98% confidence level, the critical value is approximately 2.33 (you can find this in a Z-table).

### Step 2: Plug in the values.
\[
n = \left( \frac{2.33 \cdot 300}{20} \right)^2
\]
\[
n = \left( \frac{699}{20} \right)^2 = \left( 34.95 \right)^2 \approx 1221.6
\]

Since the sample size must be a whole number, we round this up to **1222**.

### Conclusion:
The administrator should sample **1,222 students** to achieve a margin of error of 20 points at a 98% confidence level.

Would you like more details or have any questions?

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Here are five related questions to expand this topic:
1. How is the critical value for confidence intervals determined?
2. What happens to the sample size if the margin of error is decreased?
3. How does the confidence level affect the required sample size?
4. What is the difference between a Z-distribution and a t-distribution when calculating confidence intervals?
5. How does increasing the standard deviation affect the sample size?

**Tip**: Remember, as the confidence level increases, the critical value \( Z_{\alpha/2} \) increases, resulting in a larger required sample size for the same margin of error.

SAT scores are normally distributed with a mean of 1,500 and a standard deviation of 300. An administrator at a college is interested in estimating the average SAT score of first-year students. If the administrator would like to limit the margin of error of the 98% confidence interval to 20 points, how many students should the administrator sample?

Calculate the required sample size to limit the margin of error to 20 points in a 98% confidence interval, based on a normal distribution of SAT scores. Given the population standard deviation of 300, learn how to use the Z-distribution and apply statistical formulas.

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