Answer to Question #216590 in Statistics and Probability for Muhy

Question

A civic organization includes 200 members, who have an average income of $58,000, with a standard deviation of $10,000. A simple random sample of n 5 30 members is selected to participate in the annual fund-raising drive. What is the probability that the average income of the fund-raising group will be at least $60,000?

Answer 1

$n=30,$ so $\bar{X}$ is approximately normally distributed by the central limit theorem with

$$E[\bar{X}]=\mu=58000$$

$$\sigma_{\bar{X}}=\dfrac{\sigma}{\sqrt{n}}\sqrt{\dfrac{N-n}{N-1}}=\dfrac{10000}{\sqrt{30}}\sqrt{\dfrac{200-30}{200-1}}$$

$$\approx1687.4748$$

$$P(\bar{X}\geq60000)=1-P(\bar{X}<60000)$$

$$=1-P(Z<\dfrac{60000-58000}{1687.4748})$$

$$\approx1-P(Z<1.1852)\approx0.1180$$