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Math Help - The normal distribution

  1. #1
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    Question The normal distribution

    I have a Probability question from about a year ago, involving the normal distribution, which I answered but I lost the paper my answer was on, until now. I have finally found it, but now that I'm looking over my answer from a year ago, I see that there are no formulas and certainly no normal distribution table to support my answer. Would someone please be able to help me with this? I need to know what the normal distribution table will look like to support my working and where to put the formulas into my equations.

    The normal distribution
    Give clear explanations and show all working. You must provide a diagram and correct probability statements for the question.

    The weights of newborn babies in Battland are normally distributed with a mean of 3.43 kg and standard deviation of 0.36 kg.

    1% of newborn babies are classed as low birthweight babies and often in need of special care. What weight is defined as low birthweight in Battland?

    Working:
    3.43 - (0.36 x 3)
    = 2.35

    2.35 - 3.43 divided by 0.6 = 0.4987

    0.5 - 0.4987 = 0.0026

    0.5 - 0.0026 = 0.4974
    z = 2.79

    0.36 x 2.79 = 1.0044

    3.43 - 1.0044 = 2.4256
    = 2.43
    So, 2.43 kg is defined as low birthweight in Battland
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  2. #2
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    Quote Originally Posted by Kiwigirl
    I have a Probability question from about a year ago, involving the normal distribution, which I answered but I lost the paper my answer was on, until now. I have finally found it, but now that I'm looking over my answer from a year ago, I see that there are no formulas and certainly no normal distribution table to support my answer. Would someone please be able to help me with this? I need to know what the normal distribution table will look like to support my working and where to put the formulas into my equations.

    The normal distribution
    Give clear explanations and show all working. You must provide a diagram and correct probability statements for the question.

    The weights of newborn babies in Battland are normally distributed with a mean of 3.43 kg and standard deviation of 0.36 kg.

    1% of newborn babies are classed as low birthweight babies and often in need of special care. What weight is defined as low birthweight in Battland?

    Working:
    3.43 - (0.36 x 3)
    = 2.35

    2.35 - 3.43 divided by 0.6 = 0.4987

    0.5 - 0.4987 = 0.0026

    0.5 - 0.0026 = 0.4974
    z = 2.79

    0.36 x 2.79 = 1.0044

    3.43 - 1.0044 = 2.4256
    = 2.43
    So, 2.43 kg is defined as low birthweight in Battland
    Please do not make double posts on the forums.
    Attached Thumbnails Attached Thumbnails The normal distribution-picture7.gif  
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  3. #3
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    I'm really sorry about making the double post, I thought I was allowed to as long as it was in a different forum but obviously I now know I can't. But how do I make the diagram fit my answer? Also, do you know where to put the formula into my equations? Formula: z = (x - mean) divided by standard deviation.
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  4. #4
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    Quote Originally Posted by Kiwigirl
    I have a Probability question from about a year ago, involving the normal distribution, which I answered but I lost the paper my answer was on, until now. I have finally found it, but now that I'm looking over my answer from a year ago, I see that there are no formulas and certainly no normal distribution table to support my answer. Would someone please be able to help me with this? I need to know what the normal distribution table will look like to support my working and where to put the formulas into my equations.

    The normal distribution
    Give clear explanations and show all working. You must provide a diagram and correct probability statements for the question.

    The weights of newborn babies in Battland are normally distributed with a mean of 3.43 kg
    and standard deviation of 0.36 kg.

    1% of newborn babies are classed as low birthweight babies and often in need of special care. What weight is defined as low birthweight in Battland?
    As 1% of newborns are classed as low birth weight, then all babies of
    birth weight < the first percentile of a normal distribution with mean 3.43
    and standard deviation 0.36 kg are classed as of low birth weight.

    But if denotes the b is the birth weight of a child, then:

    <br />
z=\frac{b-\mbox{mean birth weight}}{\mbox{standard Dev of birth weight}}<br />

    has a standard normal distribution (as b is normally distributed),
    and a newborn is classed as of low birth weight if zis less than
    the first percentile of a standard normal distribution.

    Looking up the first percentile (0.01 value) in a table of the standard
    normal distribution gives a value of -2.32679 or about -2.33. So a baby is
    of low birth weight if:

    <br />
z=\frac{b-3.43}{0.36}<-2.33<br />

    or rearranging:

    <br />
b<(-2.33 \times 0.36)+3.43\approx 2.59\ \mbox{kg}<br />

    RonL
    Last edited by CaptainBlack; February 27th 2006 at 07:49 PM.
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  5. #5
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    Reply to CaptainBlack

    Thanks for that, you've helped me twice now, and your last explanation was excellent. I realise that I must seem dense, but probability has never been my speciality. I am still uncertain how to go about the normal distribution table and I don't quite understand your explanation this time, would you be able to simplify this so that I can have a better understanding of how to answer this with the formula? You have become a great help to me, thanks again
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  6. #6
    Grand Panjandrum
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    Quote Originally Posted by Kiwigirl
    Thanks for that, you've helped me twice now, and your last explanation was excellent. I realise that I must seem dense, but probability has never been my speciality. I am still uncertain how to go about the normal distribution table and I don't quite understand your explanation this time, would you be able to simplify this so that I can have a better understanding of how to answer this with the formula? You have become a great help to me, thanks again
    First, you need to know that if b is normally distributed with mean m, and
    standard deviation s, that:

    <br />
z=\frac{b-m}{s}<br />

    is normally distributed with mean 0 and standard deviation 1. The normal
    distribution with mean 0 and standard deviation 1 is called the standard
    normal distribution.

    Tables of the normal distribution are always for the standard normal
    distribution, which is why we transform from the meaningfully variable
    b to the slightly obscure variable z, so we can use the tabulated
    values of the standard normal distribution.

    Then a probability like: P(b<2.7)=p can we written as an equivalent
    statement about z: P(z<(2.7-m)/s)=p.

    Now the use of normal distribution table to find something like the value r such that:

    <br />
P(z<r)=0.01<br />

    can be tricky to explain, mainly because the format of the tables varies
    so much depending on the source of the table. I think I should leave
    such an explanation to a more authoritative source, like the:
    Engineering Stats Handbook

    RonL
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