To determine the number of voters that should be sampled for a [tex]\(90 \%\)[/tex] confidence interval, given that a poll reported [tex]\(42\%\)[/tex] support with a margin of error of 3.01 percentage points, we need to follow a systematic approach. Here's the step-by-step solution:
1. Identify the given parameters:
- Support percentage ([tex]\( p \)[/tex]) = [tex]\(42\%\)[/tex] = [tex]\(0.42\)[/tex] (convert percentage to a decimal)
- Margin of error ([tex]\( E \)[/tex]) = [tex]\(3.01\%\)[/tex] = [tex]\(0.0301\)[/tex] (convert percentage to a decimal)
- Confidence level: [tex]\(90\%\)[/tex]
- Critical value ([tex]\( z^* \)[/tex]) for [tex]\(90\%\)[/tex] confidence level: [tex]\(1.645\)[/tex] (from the provided table)
2. Set up the formula for the sample size (n):
The formula for the sample size needed for a given margin of error in a proportion is:
[tex]\[
n = \left( \frac{z^* \times \sqrt{p \times (1 - p)}}{E} \right)^2
\][/tex]
3. Plug in the values:
- [tex]\( z^* = 1.645 \)[/tex]
- [tex]\( p = 0.42 \)[/tex]
- [tex]\( E = 0.0301 \)[/tex]
4. Calculate the numerator:
[tex]\[
\text{Numerator} = ( z^* )^2 \times p \times (1 - p)
\][/tex]
[tex]\[
\text{Numerator} = (1.645)^2 \times 0.42 \times (1 - 0.42)
\][/tex]
[tex]\[
\text{Numerator} = 2.705625 \times 0.42 \times 0.58
\][/tex]
[tex]\[
\text{Numerator} \approx 0.65918769
\][/tex]
5. Calculate the denominator:
[tex]\[
\text{Denominator} = E^2
\][/tex]
[tex]\[
\text{Denominator} = (0.0301)^2
\][/tex]
[tex]\[
\text{Denominator} \approx 0.00090601
\][/tex]
6. Divide the numerator by the denominator to get [tex]\( n \)[/tex]:
[tex]\[
n = \frac{\text{Numerator}}{\text{Denominator}}
\][/tex]
[tex]\[
n = \frac{0.65918769}{0.00090601}
\][/tex]
[tex]\[
n \approx 727.5722011898324
\][/tex]
7. Round up to the nearest whole number:
Since we cannot sample a fraction of a person, we need to round up to the next whole number to ensure our sample size meets the required margin of error.
[tex]\[
n \approx 728
\][/tex]
Conclusion:
To achieve a [tex]\(90\%\)[/tex] confidence interval with a margin of error of 3.01 percentage points, you should sample approximately [tex]\(728\)[/tex] voters.
