Question

select the correct answer.

if \\( \hat{p} \\) is a given sample proportion, \\( n \\) is the sample size, and \\( z \\) is the number of standard deviations at a confidence level, what is the standard error of the proportion?

\\( \bigcirc \\) a. \\( \quad \sqrt{\frac{\hat{p}(\hat{p}-1)}{n}} \\)

\\( \bigcirc \\) b. \\( \quad \sqrt{\frac{\hat{p}(1-\hat{p})}{n}} \\)

\\( \bigcirc \\) c. \\( \quad n\sqrt{\hat{p}(1 - \hat{p})} \\)

\\( \bigcirc \\) d. \\( \quad z\sqrt{\frac{\hat{p}(1-\hat{p})}{n}} \\)

Explanation:

Step1: Recall the formula for the standard error of a proportion.

The standard error (SE) of a sample proportion \(\hat{p}\) is given by the formula that accounts for the sample proportion and the sample size \(n\). The formula is derived from the binomial distribution approximation (using the normal approximation for proportions), and it is \(\sqrt{\frac{\hat{p}(1 - \hat{p})}{n}}\).

Step2: Analyze each option.

• Option A: The numerator is \(\hat{p}(\hat{p}-1)\), but since \(\hat{p}\) is a proportion (between 0 and 1), \(\hat{p}-1=-(1 - \hat{p})\), and the correct numerator should be \(\hat{p}(1 - \hat{p})\), so A is incorrect.
• Option B: The formula \(\sqrt{\frac{\hat{p}(1 - \hat{p})}{n}}\) matches the formula for the standard error of a proportion, so B is correct.
• Option C: The formula \(n\sqrt{\hat{p}(1 - \hat{p})}\) has an extra factor of \(n\) and does not represent the standard error, so C is incorrect.
• Option D: The formula \(z\sqrt{\frac{\hat{p}(1 - \hat{p})}{n}}\) is the margin of error (since it multiplies the standard error by the z - score), not the standard error itself, so D is incorrect.

Answer:

B. \(\sqrt{\frac{\hat{p}(1 - \hat{p})}{n}}\)