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Math Help - Statistics - Chi-Square

  1. #1
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    Statistics - Chi-Square

    Let Xi be a random variable distributed normal with mean = i and variance = i^2.
    Assume that the random variables X1, X2 and X3 are independent.

    Using these properties of X1, X2, X3:

    a.) give a statistic formula that has a chi-square distribution with 3 degrees of freedom. - Explain.

    b.) give a statistic formula that has an F distribution with 1 and 2 degrees of freedom. - Explain.

    c.) give a statistic formula that has a t distribution with 2 degrees of freedom. - Explain.


    For part a, I did:

    [(n - 1)*
    s]/σ ~ X^2 and (n-1) degrees of freedom

    So, [3 *
    s]/
    σ would be ~ X^2 with 3 degrees of freedom if I am doing this correctly

    For part b:


    I know that the F distribution is defined as: F = (u/v1)/(w/v2) where u = chi-square with v1 degrees of freedom and w = chi-square with v2 degrees of freedom.

    I'm not really sure about how to apply it to the problem, though

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  2. #2
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    Re: Statistics - Chi-Square

    If anyone can help with this problem, it would be really appreciated.
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  3. #3
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    Re: Statistics - Chi-Square

    a) $X1^2+X2^2+X3^2$ is chi square with 3 degrees of freedom... just from the first line of the wiki definition

    b) $X1^2, X2^2, X3^2$ are all chi squared with 1 degree of freedom. $X1^2+X2^2$ for example is chi square w/2 degrees of freedom.

    so $ \dfrac{X1^2}{\dfrac{X2^2+X3^2}{2}}$ is F distributed with 1 and 2 degrees of freedom

    c) it looks like if you let $\overline{X}=\dfrac{X1+X2+X3}{3}$ then

    $\dfrac{\overline{X}-i}{ \dfrac{i^2}{ \sqrt{3}}}$ is t distributed with 2 degrees of freedom

    reading up on the t statistic makes me think it's how you answer that question about the machines being adjusted properly
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