Certainly! Let's walk through the problem step-by-step to understand how Marcus can determine the number of households he needs to survey.
1. Understanding the Given Information:
- Confidence Level: 99%
- Associated [tex]\( z \)[/tex]-score for 99% confidence: [tex]\( z^* = 2.58 \)[/tex]
- Margin of Error (E): 6% or 0.06
- Estimated Proportion [tex]\( \hat{p} \)[/tex]: Since there's no prior information, we use [tex]\( \hat{p} = 0.5 \)[/tex] (this is a standard practice to maximize the sample size).
2. Formula for Sample Size:
The formula to find the required sample size [tex]\( n \)[/tex] is:
[tex]\[
n = \hat{p}(1 - \hat{p}) \left(\frac{z^*}{E}\right)^2
\][/tex]
Here,
[tex]\[
\hat{p} = 0.5, \quad z^* = 2.58, \quad E = 0.06
\][/tex]
3. Plugging in the Values:
Let's plug these values into the formula step-by-step.
[tex]\[
n = 0.5 \times (1 - 0.5) \left(\frac{2.58}{0.06}\right)^2
\][/tex]
Simplifying further:
[tex]\[
n = 0.5 \times 0.5 \times \left(\frac{2.58}{0.06}\right)^2
\][/tex]
[tex]\[
n = 0.25 \times \left(\frac{2.58}{0.06}\right)^2
\][/tex]
[tex]\[
n = 0.25 \times \left(43\right)^2
\][/tex]
[tex]\[
n = 0.25 \times 1849
\][/tex]
[tex]\[
n = 462.25
\][/tex]
4. Rounding to the Nearest Whole Number:
Since Marcus cannot survey a fraction of a household, he must round up to the nearest whole number.
[tex]\[
n \approx 463
\][/tex]
Thus, Marcus would need to survey 463 households to meet his requirements.
From the given options, the answer is: 463 households.
