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Tyro
Asked: 2021-03-30T03:11:41+00:00 In: ,

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A certain group of welfare recipients receives benefits of $90 per week with a standard deviation of $15. If a random sample of 20 people is taken, what is the probability that their benefit will be greater than $100 per week?

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  2. acedstud
    This answer was edited.

    The probability that a  normally distributed variable, X, from a sample of size, n, with a mean, \mu, and standard deviation, \sigma, is less than a value, x is given by P(X<x)=P\left(z<\dfrac{x-\mu}{\sigma/\sqrt{n}}\right), where z is the area under the standard normal curve.

    Given: \mu=$90, \sigma=$15, n=20.

    a.) P(\bar{X}>100)=1-P(\bar{X}\le100)=1-P\left(z\le\dfrac{100-90}{15/\sqrt{20}}\right)

    =1-P\left(z\le\dfrac{10}{3.3541}\right)=1-P(z\le2.981)

    Using the table of the area under the standard normal curve or a calculator, we have that 1-P(z\le2.981)=1-0.9986=0.0014.

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