\begin{tabular}{|l|l|}
\hline
Confidence level & Critical value \\
\hline
0.90 & [tex]$z^\ \textless \ em\ \textgreater \ =1.645$[/tex] \\
\hline
0.95 & [tex]$z^\ \textless \ /em\ \textgreater \ =1.960$[/tex] \\
\hline
0.99 & [tex]$z^*=2.576$[/tex] \\
\hline
\end{tabular}

A poll reported [tex]$42 \%$[/tex] support for a statewide election with a margin of error of 3.01 percentage points.

How many voters should be sampled for a [tex]$90 \%$[/tex] confidence interval? Round up to a whole number. (Ex: 1234 voters)



Answer :

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.