To find the number of students that should be sampled to reduce the margin of error to 0.1, we can use the formula for the margin of error in a confidence interval for the mean:

$E = Z_{\alpha/2} \times \frac{\sigma}{\sqrt{n}}$

Where:

• $E$ is the margin of error.
• $Z_{\alpha/2}$ is the critical value for the desired confidence level.
• $\sigma$ is the population standard deviation.
• $n$ is the sample size.

Given:

• Desired margin of error $E = 0.1$
• Confidence level = 98%, so $Z_{\alpha/2}$ corresponds to 98% confidence.
• $\sigma = 0.78$

Step 1: Find the critical value $Z_{\alpha/2}$ for a 98% confidence level.

A 98% confidence level corresponds to $\alpha = 0.02$. The critical value $Z_{\alpha/2}$ can be found using a Z-table or standard normal distribution.

For a 98% confidence level: $Z_{\alpha/2} \approx 2.33$

Step 2: Use the formula for the margin of error to solve for $n$.

$0.1 = 2.33 \times \frac{0.78}{\sqrt{n}}$

Step 3: Solve for $n$.

$\sqrt{n} = \frac{2.33 \times 0.78}{0.1}$

$\sqrt{n} = \frac{1.8174}{0.1} = 18.174$

$n = (18.174)^2$

$n \approx 330.31$

Since $n$ must be an integer, we round up to the nearest whole number:

$n = 331$

Conclusion:

The researcher should sample 331 students to be 98% confident that the true mean is within 0.1 of the sample mean.

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Would you like more details on the steps? Here are some related questions to consider:

1. What does the margin of error represent in a confidence interval?
2. How is the critical value $Z_{\alpha/2}$ determined?
3. Why do we round up the sample size $n$ to the nearest whole number?
4. How does changing the confidence level affect the required sample size?
5. What is the impact of the population standard deviation $\sigma$ on the margin of error?
6. How would the required sample size change if the desired margin of error was smaller or larger?
7. What is the difference between a confidence interval and a prediction interval?
8. Can the margin of error be reduced without increasing the sample size?

Tip: Increasing the sample size is a common method to reduce the margin of error, but improving data collection techniques can also enhance precision without necessarily requiring a larger sample.