Answer to Question #216590 in Statistics and Probability for Muhy

Question #216590

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?



1
Expert's answer
2021-07-13T09:57:16-0400

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


E[Xˉ]=μ=58000E[\bar{X}]=\mu=58000

σXˉ=σnNnN1=1000030200302001\sigma_{\bar{X}}=\dfrac{\sigma}{\sqrt{n}}\sqrt{\dfrac{N-n}{N-1}}=\dfrac{10000}{\sqrt{30}}\sqrt{\dfrac{200-30}{200-1}}

1687.4748\approx1687.4748

P(Xˉ60000)=1P(Xˉ<60000)P(\bar{X}\geq60000)=1-P(\bar{X}<60000)

=1P(Z<60000580001687.4748)=1-P(Z<\dfrac{60000-58000}{1687.4748})

1P(Z<1.1852)0.1180\approx1-P(Z<1.1852)\approx0.1180


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