Assume that you want to construct a​ 95% confidence interval estimate of a population mean. Find an estimate of the sample size needed to obtain the specified margin of error for the​ 95% confidence interval. The sample standard deviation is given.
Margin of​ error, Eequals8.5 ​grams; sample standard​ deviation, sequals68.8 grams
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Part 1
The required sample size is 
  
enter your response here. 
​(Round up to the nearest whole​ number.)

To calculate the required sample size \( n \) for a given margin of error \( E \), we use the formula for the sample size in estimating a population mean:

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

Where:
- \( Z_{\alpha/2} \) is the z-score corresponding to the desired confidence level (for a 95% confidence interval, \( Z_{\alpha/2} = 1.96 \)).
- \( s \) is the sample standard deviation.
- \( E \) is the margin of error.

Given:
- Margin of error, \( E = 8.5 \) grams.
- Sample standard deviation, \( s = 68.8 \) grams.
- Confidence level is 95%, so \( Z_{\alpha/2} = 1.96 \).

Substitute the given values into the formula:

\[
n = \left( \frac{1.96 \cdot 68.8}{8.5} \right)^2
\]

Let's calculate that.The required sample size is 252. 

Would you like further details on the calculation or any other clarification?

Here are 5 related questions you might explore next:
1. How does changing the margin of error affect the sample size?
2. What happens to the sample size if the confidence level is increased to 99%?
3. How do you interpret a confidence interval in terms of probability?
4. What are the assumptions needed for constructing a confidence interval for a population mean?
5. How can the sample size formula change if the population standard deviation is known?

**Tip:** Increasing the confidence level increases the required sample size, as higher confidence demands more precision.

Assume that you want to construct a​ 95% confidence interval estimate of a population mean. Find an estimate of the sample size needed to obtain the specified margin of error for the​ 95% confidence interval. The sample standard deviation is given. Margin of​ error, E = 8.5 grams; sample standard​ deviation, s = 68.8 grams. (Round up to the nearest whole​ number.)

Learn how to estimate the sample size needed for a 95% confidence interval with a given margin of error of 8.5 grams and sample standard deviation of 68.8 grams. This problem covers essential statistical concepts, including confidence intervals, the sample size formula, and the Central Limit Theorem. Suitable for undergraduate statistics students.

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