Question

A national survey was initiated and ∫ended to capture the prevalence of HIV in the post-HAART (Highly Active Anti-Retroviral Therapy) era of treatment. Out of a sample of 1,483 participants, a total of 241 were found to be HIV positive. A) Calculate a SE. B) Construct a 95% confidence ∫erval about the estimate. a. A) Standard Error (SE) cannot be calculated with the given information. B) A 95% confidence ∫erval cannot be constructed with the given information. b. A) SE = (√{{p(1-p)}{n}}), B) (CI = hat{p} pm Z √{{hat{p}(1-hat{p})}{n}}) c. A) SE = (√{{n}{p(1-p)}}), B) (CI = hat{p} pm Z √{{hat{p}(1-hat{p})}{n}}) d. A) SE = (√{{p(1-p)}{n}}), B) (CI = hat{p} pm Z √{{p(1-p)}{n}})

Answer 1

A) Standard Error (SE) is (√{{n}{p(1-p)}}). B) (CI = hat{p} pm Z √{{hat{p}(1-hat{p})}{n}}). Hence, the correct answer is option C.

Step-by-step explanation:

Here's why Option C is correct.

A) Standard Error (SE):

The formula for the standard error of a proportion is:

SE = √{(p(1-p))/n}

where:

• p is the estimated proportion of HIV positive individuals (241/1483 = 0.163)
• n is the sample size (1483)

Therefore, the standard error (SE) is:

SE = √{(0.163)(1-0.163)}/1483} = 0.0112

B) 95% Confidence Interval (CI):

The formula for a 95% confidence interval for a proportion is:

CI = p ± Z √{(p(1-p))/n}

where:

• pis the estimated proportion of HIV positive individuals (0.163)
• z is the z-score corresponding to a 95% confidence level (1.96)
• n is the sample size (1483)

Therefore, the 95% confidence interval (CI) is:

CI = 0.163 ± 1.96 √{(0.163)(1-0.163)}/1483} = (0.141, 0.185)

Explanation of why other options are incorrect:

• a. A) Standard Error (SE) cannot be calculated with the given information. This is incorrect. The information provided is sufficient to calculate the standard error using the formula above.
• b. A) SE = (√{{p(1-p)}/n}), B) (CI = p pm Z √{{p(1-p)}/n}). This formula is incorrect. The denominator of the standard error formula should be the sample size (n), not 1.
• d. A) SE = (√{{p(1-p)}/n}), B) (CI = p pm Z √{{hat{p}(1-hat{p})}{n}}). This option uses the correct formula for the standard error but the wrong formula for the confidence interval. The correct formula for the confidence interval includes the estimated proportion (p) instead of the true proportion (p).

Therefore, only option c provides the correct formulas for both the standard error and the 95% confidence interval.

Correct answer: Option C