Question

A prison official wants to estimate the proportion of cases of recidivism. Examining the records of 250 convicts, the official determines that there are 65 cases of recidivism. A confidence interval will be obtained for the proportion of cases of recidivism. Part of this calculation includes the estimated standard error of the sample proportion. For this sample, the estimated standard error is

Answer 1

Answer:

For this sample, the estimated standard error is of 0.0277

Step-by-step explanation:

Central Limit Theorem

The Central Limit Theorem estabilishes that, for a normally distributed random variable X, with mean $\mu$ and standard deviation $\sigma$, the sampling distribution of the sample means with size n can be approximated to a normal distribution with mean $\mu$ and standard deviation $s = \frac{\sigma}{\sqrt{n}}$.

For a skewed variable, the Central Limit Theorem can also be applied, as long as n is at least 30.

For a proportion p in a sample of size n, the sampling distribution of the sample proportion will be approximately normal with mean $\mu = p$ and standard deviation $s = \sqrt{\frac{p(1-p)}{n}}$

Examining the records of 250 convicts, the official determines that there are 65 cases of recidivism.

This means that $n = 250, p = \frac{65}{250} = 0.26$

For this sample, the estimated standard error is

$s = \sqrt{\frac{p(1-p)}{n}} = \sqrt{\frac{0.26*0.74}{250}} = 0.0277$

For this sample, the estimated standard error is of 0.0277