If anyone can help with this problem, it would be really appreciated.
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)*s²]/σ² ~ X^2 and (n-1) degrees of freedom
So, [3 * s²]/σ² 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
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