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To solve this problem, we need to calculate the required sample size \( n \) for estimating the voter population proportion with a specified margin of error \( E \) and confidence level.

Here's the information provided:
- Margin of error \( E = 3.54\% = 0.0354 \)
- Confidence level = 90%
- Corresponding \( z \)-value for 90% confidence = 1.645

The sample size formula for estimating a population proportion is:

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

where:
- \( z \) is the critical value for the desired confidence level,
- \( E \) is the margin of error,
- \( p \) is the estimated population proportion. 

If \( p \) is unknown, we often use \( p = 0.5 \) for maximum variability (which maximizes \( p(1 - p) \)).

### Step 1: Plugging in values
Assuming \( p = 0.5 \) (since it's not provided):

\[
n = \left( \frac{1.645}{0.0354} \right)^2 \times 0.5 \times (1 - 0.5)
\]

### Step 2: Calculate the sample size
1. Calculate \( \frac{1.645}{0.0354} \):
   \[
   \frac{1.645}{0.0354} \approx 46.46
   \]

2. Square that result:
   \[
   46.46^2 \approx 2159.87
   \]

3. Multiply by \( 0.5 \times 0.5 = 0.25 \):
   \[
   n \approx 2159.87 \times 0.25 = 539.97
   \]

### Conclusion
Rounding up, the required sample size \( n \) is approximately **540**.

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

Here are some related questions to consider:
1. How would the sample size change if we used a 95% confidence level instead?
2. What if the margin of error \( E \) was reduced to 2%?
3. How does changing the estimated population proportion \( p \) impact \( n \)?
4. Why is \( p = 0.5 \) used when the true proportion is unknown?
5. What assumptions underlie the use of a normal distribution for this calculation?

**Tip:** Using a higher confidence level or a smaller margin of error generally requires a larger sample size.

A poll for a statewide election requires an error of estimation E = 3.54%. The voter count to be sampled for 90% confidence is n = ?

Calculate the required sample size for a statewide election poll with a 90% confidence interval and a 3.54% margin of error. Learn how to apply the sample size formula for population proportions, considering a critical z-value of 1.645. Ideal for college-level statistics students or advanced high school students.

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