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Student's Test Question

  1. Oct 30, 2006 #1
    Consider n+1 mutually independent random variables [itex]x+i[/itex] from a normal distribution [itex]N(\mu ,\sigma ^{2})[/itex]. Define:

    [tex] \bar{x} = \frac{1}{n} \sum_{i=1}^{n}{x_{i}} [/tex] and [tex]s^{2}=\frac{1}{n}\sum_{i=1}^{n}{\left(x_{i} - \bar{x}\right)^{2}}[/tex]

    Find the constant c so that the statistic

    [tex] t= c\frac{\bar{x} - x_{n+1}}{s} [/tex]

    follows a t-student law. Find the degrees of freedom. Justify.

    What I have so far:
    Not much, but for a statistic to follow Student T distribution with n-1:

    [tex]t=\frac{\bar{x}-\mu}{s / \sqrt{n}} [/tex]

    Because we have n+1 random variables, and s has n degrees of freedome, the resulting student t distribution will have n degree of freedoms (if s is sample standard deviation, it should have n-1 degrees of freedom). I also just expanded the above equation:

    [tex] t=c \frac {\frac{1}{n}\sum_{i=1}^{n}{x_{i}}-x_{n+1}}{\left(\frac{1}{n}\sum_{i=1}^{n}\left(x_{i} - \bar{x}\right)^{2}\right)^{\frac{1}{2}}} [/tex]

    But what next? Can someone help? thanks!!!
  2. jcsd
  3. Sep 7, 2007 #2


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    Homework Helper

    What are the two differences between the formula

    [tex] t= \frac{\bar{x} - x_{n+1}}{s} [/tex]


    [tex]t=\frac{\bar{x}-\mu}{s / \sqrt{n}} [/tex]
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