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Chi Square Distribution Problem

  1. Mar 5, 2015 #1
    1. The problem statement, all variables and given/known data
    Suppose that X has normal distribution. Find the distribution of n(Sample Mean - μ)22.
    3. The attempt at a solution
    I honestly have no idea where to begin with this problem. Any ideas?
     
  2. jcsd
  3. Mar 5, 2015 #2
    Tell us what you know about this problem. It must be homework for something you've been learning in class.
     
  4. Mar 5, 2015 #3
    Well I know that the Sample mean has a normal distribution ~ N(μ,σ2/n), which I think is useful to solve this problem. Now, I am confused about how to use this normal distribution for the sample mean to solve the problem. Any thoughts, using this idea?
     
  5. Mar 5, 2015 #4

    LCKurtz

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  6. Mar 5, 2015 #5
    Hi LCKurtz,

    Thanks for the help. I am wondering where you got this equation from? I know it's the z-score, converting the sample mean into the z-score, but how did you come up with the equation? Thanks.
     
  7. Mar 5, 2015 #6
    Actually I believe I know where this comes from.

    the Sum from i=1 to n of Zi2 = the Sum from i=1 to n of ((Xi-μ)/σ)2 = chi square distribution with n degrees of freedom.
     
  8. Mar 5, 2015 #7
    So, would the distribution of this be N(0,n)?
     
  9. Mar 5, 2015 #8

    LCKurtz

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    Now you are confusing me. It is your question asking for the distribution of$$
    \frac {n(\bar X - \mu)^2}{\sigma^2} =\frac{(\bar X - \mu)^2}{(\frac \sigma {\sqrt n})^2}$$
    So if ##Z = \frac{(\bar X - \mu)}{\frac \sigma {\sqrt n}}##, you know ##Z\sim n(0,1)## and your problem is to find the distribution of ##Z^2##.
     
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