It is known that in the absence of treatment, 68% of the patients with a certain illness will improve. The central limit theorem tells us that the percentages of patients in groups of 500 that improve in the absence of treatment are approximately normally distributed. Find the standard deviation of the normal distribution given by the central limit theorem.
A)2.1%
B)2.5%
C)3.2%
D)4.4%

Question

Quizplus · Accepted Answer

The standard deviation of the sampling distribution of a proportion is given by the formula $\sqrt{\frac{p(1-p)}{n}}$, where $p$ is the population proportion and $n$ is the sample size. Here, $p = 0.68$ and $n = 500$, so the standard deviation is $\sqrt{\frac{0.68(1-0.68)}{500}} = \sqrt{\frac{0.68 \times 0.32}{500}} = \sqrt{\frac{0.2176}{500}} = \sqrt{0.0004352} = 0.020856$, which is approximately 2.1% when converted to a percentage.

It Is Known That in the Absence of Treatment, 68