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Math Help - Approximating binomial to normal

  1. #1
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    Approximating binomial to normal

    Hi All,

    Could someone please help with the following.

    The probability that a toy balloon coming off a production line is
    faulty is 0.02. The balloons are put into bags containing 10
    balloons.
    (a)Assuming that faulty balloons occur at random, calculate the
    probability tht a bag contains a faulty balloon.
    The bags are packaged into boxes, each box containing 100 bags.
    (b)Using a suitable approximation, estimate the probability that a
    box contains 90 or more bags of fault-free balloons.

    For part (a) Use a binomial distribution with parameters of n=10 and
    p=0.02. Find the probability that X=0 and subtract it from 1 to get
    the probability that a bag contains a faulty balloon. This works out
    to be 1-0.817 = 0.183.

    For (b)Appox. to the Normal with parameter of np=100 x 0.817=81.7
    and npq=100 x 0.817 x 0.183 =14.94
    P(X>=90)=P(V>=89.5) (considering continuity correction)
    Let Z=V-81.7/(sqr-root(14.94))
    P(Z>=(89.5-81.7)/(sqr-root(14.94))
    =P(Z>=2.0173)
    =1 - P(Z<2.0173)
    =1-0.9781
    =0.0219
    However, the book is giving an answer of 0.013. Could someone please
    tell me where I have made a mistake??
    Thankyou.
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  2. #2
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    Quote Originally Posted by woollybull View Post
    Hi All,

    Could someone please help with the following.

    The probability that a toy balloon coming off a production line is
    faulty is 0.02. The balloons are put into bags containing 10
    balloons.
    (a)Assuming that faulty balloons occur at random, calculate the
    probability tht a bag contains a faulty balloon.
    The bags are packaged into boxes, each box containing 100 bags.
    (b)Using a suitable approximation, estimate the probability that a
    box contains 90 or more bags of fault-free balloons.

    For part (a) Use a binomial distribution with parameters of n=10 and
    p=0.02. Find the probability that X=0 and subtract it from 1 to get
    the probability that a bag contains a faulty balloon. This works out
    to be 1-0.817 = 0.183.

    For (b)Appox. to the Normal with parameter of np=100 x 0.817=81.7
    and npq=100 x 0.817 x 0.183 =14.94
    P(X>=90)=P(V>=89.5) (considering continuity correction)
    Let Z=V-81.7/(sqr-root(14.94))
    P(Z>=(89.5-81.7)/(sqr-root(14.94))
    =P(Z>=2.0173)
    =1 - P(Z<2.0173)
    =1-0.9781
    =0.0219
    However, the book is giving an answer of 0.013. Could someone please
    tell me where I have made a mistake??
    Thankyou.
    It looks like the book hasn't used the continuity correction .......
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