To determine the standard error of the proportion for a sample proportion [tex]\(\hat{p}\)[/tex], where [tex]\(n\)[/tex] is the sample size, we use the following formula:
[tex]\[ \text{Standard Error (SE)} = \sqrt{\frac{\hat{p}(1 - \hat{p})}{n}} \][/tex]
Let's analyze the given options to find the correct one:
A. [tex]\(\sqrt{\frac{p(p-1)}{n}}\)[/tex]
This option incorrectly uses [tex]\(p(p-1)\)[/tex] instead of [tex]\(\hat{p}(1 - \hat{p})\)[/tex], making it incorrect.
B. [tex]\(\sqrt{\frac{\dot{p}(1-\hat{p})}{n}}\)[/tex]
This option correctly follows the form of the standard error formula, [tex]\(\sqrt{\frac{\hat{p}(1 - \hat{p})}{n}}\)[/tex], but the notation [tex]\(\hat{p}\)[/tex] should be used consistently. Nevertheless, it represents the correct mathematical formulation.
C. [tex]\(n \sqrt{\hat{p}(1-\hat{p})}\)[/tex]
This option includes a factor of [tex]\(n\)[/tex] inside the square root, which is not part of the standard error formula. Thus, this is incorrect.
D. [tex]\(z \sqrt{\frac{\hat{p}(1-\dot{p})}{n}}\)[/tex]
This option incorrectly introduces [tex]\(z\)[/tex] as a multipler, which is not part of the standard error formula. This is incorrect as well.
Given these choices, the correct answer is option B: [tex]\(\sqrt{\frac{\hat{p}(1-\hat{p})}{n}}\)[/tex].
