Estimate the minimum sample size needed to achieve the margin of error Eequals0.009 for a​ 95% confidence interval.
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Part 1
The minimum sample size is 
  
enter your response here. ​(Round up to the nearest​ integer.)

To estimate the minimum sample size needed for a 95% confidence interval with a given margin of error \(E = 0.009\), we can use the formula for sample size in the context of estimating a population proportion:

\[
n = \left( \frac{Z_{\alpha/2}^2 \cdot p \cdot (1 - p)}{E^2} \right)
\]

Where:
- \(n\) is the sample size.
- \(Z_{\alpha/2}\) is the Z-score corresponding to the desired confidence level.
- \(p\) is the estimated population proportion (if unknown, use \(p = 0.5\) to maximize the sample size).
- \(E\) is the margin of error.

For a 95% confidence level, the Z-score \(Z_{\alpha/2}\) is approximately 1.96.

### Steps:

1. Assume \(p = 0.5\) (as this will give the maximum sample size).
2. Plug in the values:

\[
n = \left( \frac{(1.96)^2 \cdot 0.5 \cdot (1 - 0.5)}{(0.009)^2} \right)
\]

Let's calculate the value of \(n\).The minimum sample size required to achieve a margin of error of 0.009 for a 95% confidence interval is **11,857**.

Would you like more details on this process or have any questions?

Here are 5 related questions to expand on this concept:
1. How does changing the margin of error affect the sample size required?
2. What is the formula for calculating the Z-score for different confidence levels?
3. How would the sample size change if the confidence level was 90% instead of 95%?
4. What happens if the population proportion \(p\) is known to be something other than 0.5?
5. How does the sample size calculation differ for continuous data vs. proportions?

**Tip:** Always round up when calculating sample size, as partial samples don't exist in practice.

Estimate the minimum sample size needed to achieve the margin of error E = 0.009 for a 95% confidence interval.

Learn how to calculate the minimum sample size needed to achieve a margin of error of 0.009 for a 95% confidence interval. This guide covers key concepts in statistics, such as confidence intervals, Z-scores, and sample size formulas. Suitable for undergraduate students.

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