To determine the minimum sample size required for the estimate of [tex]\(\hat{p}\)[/tex] (proportion) to be within a margin of error [tex]\( E \)[/tex] at a 95% confidence level, we use the following formula:
[tex]\[ n = \hat{p}(1 - \hat{p}) \left(\frac{z^}{E}\right)^2 \][/tex]
Where:
- [tex]\(\hat{p} = 0.58\)[/tex] is the point estimate of the population proportion.
- [tex]\(z^ = 1.96\)[/tex] is the z-score corresponding to a 95% confidence level.
- [tex]\(E = 0.04\)[/tex] is the margin of error.
Let's break down the calculation step-by-step:
1. Calculate [tex]\( \hat{p}(1 - \hat{p}) \)[/tex]:
[tex]\[ \hat{p}(1 - \hat{p}) = 0.58 \times (1 - 0.58) \][/tex]
[tex]\[ = 0.58 \times 0.42 \][/tex]
[tex]\[ = 0.2436 \][/tex]
2. Calculate [tex]\( \left(\frac{z^*}{E}\right)^2 \)[/tex]:
[tex]\[ \left(\frac{1.96}{0.04}\right)^2 \][/tex]
[tex]\[ = \left(\frac{1.96}{0.04}\right) \times \left(\frac{1.96}{0.04}\right) \][/tex]
[tex]\[ = 49^2 \][/tex]
[tex]\[ = 2401 \][/tex]
3. Multiply the results of step 1 and step 2:
[tex]\[ n = 0.2436 \times 2401 \][/tex]
[tex]\[ = 0.9358137599999999 \][/tex]
4. Divide by the square of the margin of error:
[tex]\[ \text{Margin of Error squared} = 0.04^2 \][/tex]
[tex]\[ = 0.0016 \][/tex]
[tex]\[ \text{sample size} = \frac{0.9358137599999999}{0.0016} \][/tex]
[tex]\[ = 584.8835999999999 \][/tex]
Since the sample size must be a whole number, we round up to the nearest whole number:
[tex]\[ \text{Minimum sample size} = 585 \][/tex]
Thus, the minimum sample size that should be used so that the estimate of [tex]\(\hat{p}\)[/tex] will be within the required margin of error of the population proportion is 585.
So, the correct answer is:
[tex]\[ \boxed{585} \][/tex]
