A population has standard deviation 0=17.2. A sample size of 
What is needed to be drawn in order to obtain a  confidence interval with a margin of error equal to 3.8? 

To determine the sample size needed for a specified margin of error in a confidence interval, we can use the formula:

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

where:
- \( n \) is the sample size,
- \( Z \) is the Z-score corresponding to the desired confidence level,
- \( \sigma \) is the population standard deviation,
- \( E \) is the margin of error.

Given:
- \( \sigma = 17.2 \)
- \( E = 3.8 \)

Assuming you want a 95% confidence interval (which corresponds to a Z-score of approximately 1.96), the calculation would be:

\[
n = \left( \frac{1.96 \cdot 17.2}{3.8} \right)^2
\]

Now, let's calculate it step by step.

1. Calculate \( \frac{1.96 \cdot 17.2}{3.8} \):
   \[
   \frac{1.96 \cdot 17.2}{3.8} = \frac{33.712}{3.8} \approx 8.86
   \]

2. Square the result:
   \[
   n \approx (8.86)^2 \approx 78.5
   \]

Since the sample size must be a whole number, we round up to ensure we meet the margin of error. Thus, the required sample size \( n \) is approximately **79**.

### Summary
To achieve a margin of error of 3.8 with a standard deviation of 17.2 at a 95% confidence level, a sample size of **79** is needed.

Would you like more details or have any questions? Here are some related questions you might consider:

1. What is the effect of increasing the confidence level on the required sample size?
2. How does the population standard deviation impact the sample size needed?
3. What happens to the margin of error if the sample size increases?
4. Can you explain the difference between a population and a sample in statistics?
5. How would you calculate the sample size for different confidence levels (e.g., 90% or 99%)? 

**Tip:** Always round up your sample size to ensure the margin of error is met.

A population has standard deviation 0=17.2. A sample size of what is needed to be drawn in order to obtain a confidence interval with a margin of error equal to 3.8?

Discover how to calculate the required sample size for a confidence interval with a margin of error of 3.8 and a population standard deviation of 17.2. This guide provides a step-by-step solution using the sample size formula and discusses the impact of confidence levels.

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