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Math Help - need to solve summation equation to solve sum(x2)

  1. #1
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    need to solve summation equation to solve sum(x2)

    So I need to calculate an estimated variance, the info given to me is:
    A sample of 140 bags of flour. The masses of x grams of the contents are summarized by \sum (x - 500) = -266 and  \sum (x-500)^2=1178 I need to find the mean and estimated variance. The mean is simple 140(x - 500) = -266; mean = 498.3 But how the heck do I figure out \sum x^2

    Someone suggested :
    <br />
\sum_{i = 1}^{140}(x_i - 500)^2 = 1178<br />
    <br />
\Rightarrow \sum_{i = 1}^{140}x_i^2 -2\sum_{i = 1}^{140} 500*x_i + \sum_{i = 1}^{140}500^2 = 1178<br />

    But unfortunately I dont even know how to solve the above equations, I did google and read this : A-level Mathematics/FP1/Summation of Series - Wikibooks, collection of open-content textbooks

    But It doesnt apply here since we're dealing with a random variable xi? Right? How do I simplify the second and third summations on the left

    I dont need the answer, I just a hint or maybe how to solve summations like this? Could someone point me in the right direction?
    (Im studying for my A levels on my own =S)
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  2. #2
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    Quote Originally Posted by giddy View Post
    So I need to calculate an estimated variance, the info given to me is:
    A sample of 140 bags of flour. The masses of x grams of the contents are summarized by \sum (x - 500) = -266 and  \sum (x-500)^2=1178 I need to find the mean and estimated variance. The mean is simple 140(x - 500) = -266; mean = 498.3 But how the heck do I figure out \sum x^2

    Someone suggested :
    <br />
\sum_{i = 1}^{140}(x_i - 500)^2 = 1178<br />
    <br />
\Rightarrow \sum_{i = 1}^{140}x_i^2 -2\sum_{i = 1}^{140} 500*x_i + \sum_{i = 1}^{140}500^2 = 1178<br />

    But unfortunately I dont even know how to solve the above equations, I did google and read this : A-level Mathematics/FP1/Summation of Series - Wikibooks, collection of open-content textbooks

    But It doesnt apply here since we're dealing with a random variable xi? Right? How do I simplify the second and third summations on the left

    I dont need the answer, I just a hint or maybe how to solve summations like this? Could someone point me in the right direction?
    (Im studying for my A levels on my own =S)
    You know that:

    \displaystyle {-266 = \sum_{i = 1}^{140}(x_i - 500) = \sum_{i = 1}^{140} x_i - (500)(140) = \sum_{i = 1}^{140} x_i -70000 \Rightarrow \sum_{i = 1}^{140} x_i = .....}


    You also know that \displaystyle {1178 = \sum_{i = 1}^{140}(x_i - 500)^2 = \sum_{i = 1}^{140} x_i^2 - 1000 \sum_{i = 1}^{140} x_i + 500^2 (140) }.

    Substitute the value of \displaystyle {\sum_{i = 1}^{140} x_i} and make \displaystyle {\sum_{i = 1}^{140} x_i^2} the subject.


    Note: \displaystyle {\sum_{i = 1}^{140} 1 = 140 (1) = 140 ....}
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  3. #3
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    Smile

    Darn! That was simple I just didn't see it! =S

    Thanks so much! On my way now to hypothesis testing of discrete variables!!
    Last edited by mr fantastic; July 17th 2010 at 12:22 AM.
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