In a random sample of 31 professional actors, it was found that 24 were extroverts. Find a 98% confidence interval for p.
A)0.589 to 0.959
B)0.647 to 1.485
C)0.647 to 0.902
D)0.589 to 1.485
E)0.063 to 0.959

Question

Quizplus · Accepted Answer

The confidence interval for the proportion $p$ of extroverts among professional actors can be calculated using the formula for a confidence interval for a proportion: $\hat{p} \pm Z_{\alpha/2} \sqrt{\frac{\hat{p}(1-\hat{p})}{n}}$, where $\hat{p} = \frac{x}{n}$ (x is the number of successes, n is the sample size), and $Z_{\alpha/2}$ is the Z-score corresponding to the desired confidence level. Here, $\hat{p} = \frac{24}{31}$, $n = 31$, and for a 98% confidence interval, $Z_{\alpha/2}$ is approximately 2.33 (since the Z-score for 99% confidence level in one tail is 2.33, making it 98% for two tails). Plugging these values into the formula gives a confidence interval that, when calculated, falls within the range provided in option A, 0.589 to 0.959.

In a Random Sample of 31 Professional Actors, It Was