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dnlllzld
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Tyro
Asked: 2021-03-30T05:37:22+00:00 In: ,

Statistics and probability

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Assume that women’s heights are normally distributed with a mean 64.5 in, and a standard deviation of 2.3 in.

a. if 1 woman is randomly selected, find the probability that her height is less than 65 in.

b. if 46 women are randomly selected, find the probability that they have a mean height less than 65 in.

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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=64.5 in, \sigma=2.3 in

    a.) P(X<65)=P\left(z<\dfrac{65-64.5}{2.3/\sqrt{1}}\right)

    =P\left(z<\dfrac{0.5}{2.3}\right)=P(z<0.217)

    Using the table of the area under the standard normal curve or a calculator, we have that P(z<0.217)=0.5861.

     

    b.) P(\bar{X}<65)=P\left(z<\dfrac{65-64.5}{2.3/\sqrt{46}}\right)

    =P\left(z<\dfrac{0.5}{0.339116}\right)=P(z<1.474)

    Using the table of the area under the standard normal curve or a calculator, we have that P(z<1.474)=0.9298.

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