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To solve this problem, we need to find the sample size necessary to estimate the true proportion (42% or 0.42) of registered voters who agree with the politician's view, within a margin of error of 7% (or 0.07), with 80% confidence.

We can use the following formula for sample size in proportion problems:

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

Where:
- \( Z \) is the z-score corresponding to the desired confidence level.
- \( p \) is the estimated proportion of success (0.42).
- \( E \) is the margin of error (0.07).
- \( n \) is the required sample size.

### Step 1: Find the Z-score
For an 80% confidence level, the z-score is approximately **1.28** (this comes from standard normal distribution tables).

### Step 2: Apply the formula

\[
n = \left(\frac{1.28^2 \cdot 0.42 \cdot (1 - 0.42)}{0.07^2}\right)
\]

\[
n = \left(\frac{1.6384 \cdot 0.42 \cdot 0.58}{0.0049}\right)
\]

\[
n = \left(\frac{1.6384 \cdot 0.2436}{0.0049}\right)
\]

\[
n = \left(\frac{0.3989}{0.0049}\right)
\]

\[
n \approx 81.41
\]

### Step 3: Round to the nearest whole number
The sample size required is approximately **81**.

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Would you like further details on how the formula works or have any additional questions?

### Related Questions:
1. What is the margin of error, and how does it affect the sample size?
2. Why do we use the z-score for determining sample sizes in confidence intervals?
3. How does increasing the confidence level affect the required sample size?
4. What is the relationship between the proportion \( p \) and sample size?
5. What happens if the margin of error is reduced to 5% in this problem?

### Tip:
As the confidence level increases, the required sample size also increases because a higher confidence level demands more precision in estimation.

The public relations manager for a political candidate would like to determine if the 42% of the registered voters in the candidate's district agree with the politician’s view on a particular local issue. Find the sample size necessary for the public relations manager to estimate the true proportion to within 7% with 80% confidence. (Round to the nearest whole number).

This example demonstrates how to calculate the sample size needed to estimate a true proportion with 80% confidence and a 7% margin of error. The problem involves determining if 42% of registered voters agree with a political candidate's stance. Learn about key statistical concepts such as confidence intervals, margin of error, and sample size calculations.

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