I am trying to calculate upper and lower confidence levels for a parameter, but i can't get it straight (in this case σ 2 ):the reference variable: R σ 2 := n − 1 s 2 σ 2 ∼ χ 2 ( n − 1 )where s 2 = 1 n − 1 ∑ i = 1 n ( x i − x ¯ ) 2

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I am trying to calculate upper and lower confidence levels for a parameter, but i can&#039;t get it straight (in this case       σ    2   ):the reference variable:       R                  σ        2              :=            n      −      1              s        2                    σ      2        ∼      χ    2    (  n  −  1  )where       s    2    =      1          n      −      1            ∑          i      =      1              n        (      x    i    −            x      ¯            )    2

Krista Leon · Accepted Answer

Step 1  α  =  P  (      R                  σ        2              &amp;gt;      χ          α      ,      n      −      1        2    )  =  P  (            (      n      −      1      )              s        2                    σ      2        &amp;gt;      χ          α      ,      n      −      1        2    )  =  P  (            σ      2              (      n      −      1      )              s        2              &amp;gt;      1          χ              1        −        α        ,        n        −        1            2        )  =  P  (      σ    2    &amp;gt;            (      n      −      1      )              s        2                    χ              1        −        α        ,        n        −        1            2        )    ⟹                      σ        2            ¯        =            (      n      −      1      )              s        2                    χ              1        −        α        ,        n        −        1            2        .A good way to check your answer is that you know when   α becomes small,       χ          α      ,      n      −      1        2   becomes large and       χ          α      ,      n      −      1        2   becomes small for any value of   n  &amp;gt;  1 . This means that             1              χ                  1          −          α          ,          n          −          1                2             becomes increasingly large as   α  →  0  , , so you know that             1              χ                  1          −          α          ,          n          −          1                2             is the upper confidence level.

I am trying to calculate upper and lower confidence levels for a parameter, but i can't get it straight (in this case sigma^2): the reference variable: R_(sigma^2):=(n-1_(s^2))/(sigma^2)~chi^2(n-1) where s^2=1/(n-1)sum_(i=1)^n(x_i-bar(x))^2

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