You want to estimate the proportion of voters who will vote for Candidate X. From prior polls you know that the population proportion is no larger than 0.40 (that is, Candidate X has never polled higher than 40%). If you want a 95% confidence interval estimate of the true population proportion, with a margin of error no greater than 0.03, what minimum sample size is required?

To find the minimum sample size required to estimate the proportion of voters who will vote for Candidate X, we can use the formula for determining the sample size for a population proportion:

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

Where:
- \( n \) is the required sample size.
- \( Z \) is the Z-score corresponding to the desired confidence level.
- \( p \) is the estimated population proportion.
- \( E \) is the margin of error.

### Step 1: Identify the known values.
- Confidence level is 95%, so the Z-score is approximately \( 1.96 \) (based on the standard normal distribution).
- From the problem, the estimated population proportion \( p \leq 0.40 \). To ensure the maximum sample size, we use \( p = 0.40 \).
- The margin of error \( E = 0.03 \).

### Step 2: Plug the values into the formula.

\[
n = \frac{(1.96)^2 \cdot 0.40 \cdot (1 - 0.40)}{(0.03)^2}
\]

\[
n = \frac{(1.96)^2 \cdot 0.40 \cdot 0.60}{0.0009}
\]

### Step 3: Calculate the sample size.

\[
n = \frac{3.8416 \cdot 0.24}{0.0009}
\]

\[
n = \frac{0.922}{0.0009}
\]

\[
n \approx 1024.44
\]

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

\[
n = 1025
\]

### Conclusion:
The minimum sample size required to estimate the proportion of voters for Candidate X with a 95% confidence level and a margin of error of 0.03 is **1025**.

Would you like further clarification or details on any part of this process?

### Related Questions:
1. How would the sample size change if the confidence level were increased to 99%?
2. What happens to the sample size if the margin of error is reduced to 0.02?
3. How is the sample size affected if the population proportion is lower, say \( p = 0.30 \)?
4. What is the formula to estimate the sample size when the population proportion is unknown?
5. How can you interpret the confidence interval once the sample data is collected?

### Tip:
To minimize the required sample size, adjusting the margin of error or confidence level can significantly impact the result. Keep this in mind when designing surveys or polls.

Learn how to calculate the minimum sample size needed to estimate the population proportion of voters for Candidate X. This guide uses a 95% confidence level, a margin of error of 0.03, and an estimated proportion of 0.40 to show step-by-step how to apply the formula n = (Z^2 * p * (1 - p)) / E^2.

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