Let's complete the statements step-by-step:
1. Identify the sample size:
The sample size is the number of employees polled.
- The sample size in this problem is [tex]\(500\)[/tex] employees.
2. Calculate the population proportion:
The population proportion is the fraction of the sample who support raising the minimum wage.
- The population proportion is estimated as [tex]\(\frac{435}{500} = 0.87\)[/tex].
3. Determine the [tex]\(z^\)[/tex]-score:
The [tex]\(z^\)[/tex]-score for a 95% confidence level is given as 1.96.
4. Calculate the margin of error using the formula:
The formula for the margin of error [tex]\(E\)[/tex] is:
[tex]\[
E = z^* \cdot \sqrt{\frac{\hat{p}(1 - \hat{p})}{n}}
\][/tex]
where:
- [tex]\(\hat{p}\)[/tex] is the population proportion (0.87),
- [tex]\(n\)[/tex] is the sample size (500),
- and [tex]\(z^*\)[/tex] is the z-score (1.96).
5. Substitute the values into the formula:
[tex]\[
E = 1.96 \cdot \sqrt{\frac{0.87 \cdot (1 - 0.87)}{500}}
\][/tex]
6. Simplify the expression:
[tex]\[
E = 1.96 \cdot \sqrt{\frac{0.87 \cdot 0.13}{500}}
\][/tex]
[tex]\[
E = 1.96 \cdot \sqrt{\frac{0.1131}{500}}
\][/tex]
[tex]\[
E = 1.96 \cdot \sqrt{0.0002262}
\][/tex]
[tex]\[
E = 1.96 \cdot 0.01504 \approx 0.0295
\][/tex]
7. Convert to a percentage and round to the nearest tenth:
To express the margin of error as a percentage, multiply by 100.
[tex]\[
0.0295 \times 100 = 2.95\%
\][/tex]
Rounding to the nearest tenth of a percent gives us [tex]\(2.9\%\)[/tex].
Therefore, the completed statements are:
- The sample size in this problem is [tex]\(500\)[/tex] employees.
- The population proportion is estimated as [tex]\(0.87\)[/tex].
- When the margin of error is calculated using the formula [tex]\(E = z^* \cdot \sqrt{\frac{\hat{p}(1 - \hat{p})}{n}}\)[/tex], to the nearest tenth of a percent, the result is [tex]\(2.9\%\)[/tex].
