Given the parameters from the question, we need to determine the minimum sample size [tex]\( n \)[/tex] such that the estimate of [tex]\(\hat{p}\)[/tex] will be within the specified margin of error [tex]\((E)\)[/tex].
We are given:
- Sample proportion [tex]\(\hat{p} = 0.34\)[/tex]
- Margin of error [tex]\(E = 0.027\)[/tex]
- [tex]\(z^\)[/tex]-score corresponding to a 90% confidence level [tex]\(z^ = 1.645\)[/tex]
The formula to determine the minimum sample size is:
[tex]\[ n = \hat{p}(1 - \hat{p}) \left(\frac{z^}{E}\right)^2 \][/tex]
Now, substituting the given values into the formula:
1. [tex]\(\hat{p} = 0.34\)[/tex]
2. [tex]\(1 - \hat{p} = 1 - 0.34 = 0.66\)[/tex]
3. [tex]\(\frac{z^}{E} = \frac{1.645}{0.027} \approx 60.926\)[/tex]
Next, calculate the expression inside the formula:
[tex]\[ \left(\frac{1.645}{0.027}\right)^2 \approx 3700.184 \][/tex]
Then, multiply by [tex]\(\hat{p}(1 - \hat{p})\)[/tex]:
[tex]\[ n = 0.34 \times 0.66 \times 3700.184 \approx 832.966 \][/tex]
Thus, the minimum sample size needed is approximately 833.
Therefore, the correct answer is:
[tex]\[ \boxed{833} \][/tex]
