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Thread: Applications of Normal Distributions

  1. #1
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    Applications of Normal Distributions

    Happy Saturday!

    I am getting a slightly different answer than the answer key for this problem, and I don't know why? The problem is:

    Men's heights are normally distributed with mean 69.5 inches and standard deviation 2.4 inches.
    The US Navy requires that fighter pilots have heights between 62 inches and 78 inches.
    1. Find the percentage of men meeting the height requirement.
    2. If the Navy changes the height requirements so that all men are eligible except the shortest 1% and the tallest 1%, what are the new height requirements for men?

    The answer key has 1. 94.89% and 2. 63.91 to 75.08 inches. Can someone explain so I can see where I'm going wrong?
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  2. #2
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    Re: Applications of Normal Distributions

    Let $\Phi(x)$ be the CDF of the standard normal distribution, i.e. zero mean, unit variance.

    1) $ p = \Phi\left(\dfrac{78-69.5}{2.4}\right) - \Phi\left(\dfrac{62-69.5}{2.4}\right) \approx 0.9989 = 99.89\%$

    The answer key is wrong or you have given us wrong information.

    2) we want to find $hi, ~lo$ such that

    $\Phi\left(\dfrac{hi-69.5}{2.4}\right) = 0.99$

    and

    $\Phi\left(\dfrac{lo-69.5}{2.4}\right) = 0.01$

    using a table or software to look up the z-scores for the values to the right of the equals sign we get

    $\dfrac{hi-69.5}{2.4} = 2.32635$

    $hi \approx 75.08~in$


    $\Phi\left(\dfrac{lo-69.5}{2.4}\right) = -2.32635$

    $lo \approx 63.92~in$
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