My Math Forum Confidence Intervals...

 November 14th, 2012, 02:39 PM #1 Member   Joined: May 2012 Posts: 56 Thanks: 0 Confidence Intervals... Let X1, X2, ... , Xn be a random sample from $N(\mu, \sigma^2)$, where both parameters $\mu$ and $\sigma^2$ are unknown. A confidence interval for $\sigma^2$ can be found as follows. We know that $(n-1)S^2/\sigma^2$ is a random varible with $X^2(n-1)$ distribution. Thus we can find constants a and b so that $P((n-1)S^2/\sigma^2=< b)= 0.975$ and $P(a=< (n-1)S^2/\sigma^2=< b)=0.95$. a) Show that this second probability statement can be written as $P((n-1)S^2/b=< \sigma^2=< (n-1)S^2/a)= 0.95$. I could do this by flipping all of them, changing the signs...and then mulitplying all of them by (n-1)S^2. b) If n=9 adn s^2 = 7.93, find a 95% confidence interval for $\sigma^2$. Here, I just substitute n=9 and s^2=7.93 to the formula, right? c) If $\mu$ is known, how would you modify the preceding procedure for finding a confidence interval for $\sigma^2$. I am confused with this one...so can anybody give me a hint or something? Thanks in advance

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