A survey was given to a random sample of 50 residents of a town to determine whether they support a new plan to raise taxes in order to increase education spending. Of those surveyed, 20 respondents said they were in favor of the plan. At the 95% confidence level, what is the margin of error for this survey expressed as a proportion to the nearest thousandth? (Do not write plus or minus±).

To find the margin of error for the survey at the 95% confidence level, we can use the formula for the margin of error (ME) in a proportion:

\[
\text{ME} = z \times \sqrt{\frac{p(1 - p)}{n}}
\]

Where:
- \( z \) is the z-score corresponding to the desired confidence level (for 95% confidence, \( z \approx 1.96 \)),
- \( p \) is the sample proportion,
- \( n \) is the sample size.

Given:
- Sample size \( n = 50 \),
- Number of favorable responses \( x = 20 \),
- Proportion \( p = \frac{20}{50} = 0.4 \).

Let's calculate the margin of error:

\[
\text{ME} = 1.96 \times \sqrt{\frac{0.4 \times (1 - 0.4)}{50}}
\]

\[
\text{ME} = 1.96 \times \sqrt{\frac{0.4 \times 0.6}{50}}
\]

\[
\text{ME} = 1.96 \times \sqrt{\frac{0.24}{50}}
\]

\[
\text{ME} = 1.96 \times \sqrt{0.0048}
\]

\[
\text{ME} = 1.96 \times 0.0693
\]

\[
\text{ME} \approx 0.135
\]

So, the margin of error for this survey at the 95% confidence level is approximately **0.135**.

Would you like to go through the steps in more detail or ask about related concepts? Here are some questions that might be useful:

1. What is the significance of the 95% confidence level?
2. How does the sample size affect the margin of error?
3. What would happen to the margin of error if the sample proportion \( p \) were closer to 0 or 1?
4. How is the z-score determined for different confidence levels?
5. What would the margin of error be for a different sample size?

**Tip:** The margin of error decreases as the sample size increases, which means larger samples provide more precise estimates.

Learn how to calculate the margin of error for a survey conducted to gauge support for a tax increase in a town. This problem involves key statistical concepts such as the margin of error and confidence intervals, using a sample of 50 residents. Suitable for students in Grades 10-12.

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