What sample size should be obtained if we want to be within four percentage points with 96% confidence if we use an estimate of 48% obtained from a poll?

To determine the sample size required to achieve a specified margin of error with a given confidence level, we can use the formula for the sample size \( n \) for estimating a population proportion. The formula is:

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

where:
- \( Z \) is the Z-score corresponding to the desired confidence level,
- \( p \) is the estimated proportion,
- \( E \) is the margin of error.

Given:
- Confidence level: 96%
- Margin of error: 4% (0.04)
- Estimated proportion (\( p \)): 48% (0.48)

First, find the Z-score for a 96% confidence level. For a 96% confidence level, the Z-score corresponds to the area in the tails of the normal distribution such that 2% is in each tail (100% - 96% = 4%, so each tail has 2%).

Using a Z-table or standard normal distribution table, the Z-score for 96% confidence is approximately 2.05.

Now, plug these values into the formula:

\[ n = \left( \frac{2.05^2 \cdot 0.48 \cdot (1 - 0.48)}{0.04^2} \right) \]

Calculate the components step by step:

1. \( 2.05^2 = 4.2025 \)
2. \( 0.48 \cdot (1 - 0.48) = 0.48 \cdot 0.52 = 0.2496 \)
3. \( 0.04^2 = 0.0016 \)

Now, compute the sample size:

\[ n = \left( \frac{4.2025 \cdot 0.2496}{0.0016} \right) \]

\[ n = \left( \frac{1.049452}{0.0016} \right) \]

\[ n = 655.9075 \]

Since the sample size must be a whole number, round up to the next whole number:

\[ n \approx 656 \]

Therefore, the sample size required is 656.

Would you like more details or have any questions about this calculation?

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Here are 8 relative questions that expand on this information:

1. What is the Z-score for a 95% confidence level?
2. How does the margin of error affect the sample size?
3. What happens to the sample size if the estimated proportion \( p \) is closer to 0.5?
4. How can we determine the Z-score without a Z-table?
5. What is the formula for determining sample size for a population mean?
6. How does increasing the confidence level affect the required sample size?
7. What is the impact of a smaller margin of error on the sample size?
8. How would the sample size change if the confidence level was increased to 99%?

**Tip:** Always round up the calculated sample size to ensure the margin of error is not exceeded.

Learn how to calculate the sample size required to estimate a population proportion with 96% confidence level and 4% margin of error. Detailed step-by-step solution using statistical formulas and normal distribution properties.

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