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Thread: Estimate mean and std dev?

  1. #1
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    Cool Estimate mean and std dev?

    A random sample of 13 observations from a population yielded $\displaystyle \sum$ x=488.8 and $\displaystyle \sum$ $\displaystyle x^2$ = 18950.2. Estimate $\displaystyle \mu$ and $\displaystyle \sigma.$

    I can't find this in my notes, I think $\displaystyle \sum$ x = 488.8 would be the sum of the sample, not sure where to go from there.

    Thank you
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  2. #2
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    Quote Originally Posted by rba_mandy View Post
    A random sample of 13 observations from a population yielded $\displaystyle \sum$ x=488.8 and $\displaystyle \sum$ $\displaystyle x^2$ = 18950.2. Estimate $\displaystyle \mu$ and $\displaystyle \sigma.$

    I can't find this in my notes, I think $\displaystyle \sum$ x = 488.8 would be the sum of the sample, not sure where to go from there.

    Thank you
    $\displaystyle \overline{x} = E(X) = \frac{\sum_{i=1}^n x_i}{n}$


    $\displaystyle s_X^2 = Var(X) = \frac{\sum_{i=1}^n (x_i - \overline{x})^2}{n} = \frac{\left(\sum_{i=1}^n x_i^2\right) - 2 \overline{x} \left( \sum_{i=1}^n x_i \right) + n \overline{x}^2}{n}$.


    $\displaystyle = \frac{\sum_{i=1}^n x_i^2}{n} - 2 \overline{x} \frac{\sum_{i=1}^n x_i}{n} + \overline{x}^2$


    $\displaystyle = \frac{\sum_{i=1}^n x_i^2}{n} - \overline{x}^2$.


    Substitute your data and do the calculations.
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  3. #3
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    Quote Originally Posted by mr fantastic View Post
    $\displaystyle \overline{x} = E(X) = \frac{\sum_{i=1}^n x_i}{n}$


    $\displaystyle s_X^2 = Var(X) = \frac{\sum_{i=1}^n (x_i - \overline{x})^2}{n} = \frac{\left(\sum_{i=1}^n x_i^2\right) - 2 \overline{x} \left( \sum_{i=1}^n x_i \right) + n \overline{x}^2}{n}$.


    $\displaystyle = \frac{\sum_{i=1}^n x_i^2}{n} - 2 \overline{x} \frac{\sum_{i=1}^n x_i}{n} + \overline{x}^2$


    $\displaystyle = \frac{\sum_{i=1}^n x_i^2}{n} - \overline{x}^2$.


    Substitute your data and do the calculations.

    Ok, but I don't understand these formulas. What is $\displaystyle \frac{\sum_{i=1}^n x_i}{n}$ the n over i?

    Is the first formula for $\displaystyle \mu$ and the second for $\displaystyle \sigma$?

    Could you possibly give me a head start or an example maybe?

    Thanks for your help.
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  4. #4
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    Quote Originally Posted by rba_mandy View Post
    Ok, but I don't understand these formulas. What is $\displaystyle \frac{\sum_{i=1}^n x_i}{n}$ the n over i?

    Is the first formula for $\displaystyle \mu$ and the second for $\displaystyle \sigma$?

    Could you possibly give me a head start or an example maybe?

    Thanks for your help.
    $\displaystyle \sum_{i=1}^n x_i$ is the sum of all the data.

    $\displaystyle \sum_{i=1}^n x_i^2$ is the sum of the squares of all the data.

    $\displaystyle n$ is the number of data points.

    You have been given the value of these three things. Substitute them into the formulae I gave you.
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  5. #5
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    Smile

    Ok, I think I understand. Thank you for the help.
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