Solution :
a). Given :
a = [tex]$0.1$[/tex],
chisquare with the [tex]$0.05$[/tex],
[tex]$df=11$[/tex] is [tex]$4.57$[/tex] (from the chisquare table)
Chisquare with the [tex]$0.95$[/tex], df = [tex]$11$[/tex] is [tex]$19.68$[/tex] (from chisquare table)
So 90% Cl is
[tex]$\left(\frac{v(n-1)\times s}{v19.68}, \frac{v(n-1)\times s}{v4.57}\right)$[/tex]
[tex]$=\frac{\sqrt{11}\times 147.928}{\sqrt{19.68}},\frac{\sqrt{11}\times 147.928}{\sqrt{4.57}}$[/tex]
[tex]$=(110.5947,229.5031)$[/tex]
b). Given :
a = [tex]$0.05$[/tex]
Chisquare with [tex]$0.025$[/tex]
df = [tex]$n-1=11$[/tex] is A, [tex]$3.82$[/tex]
Chisquare with [tex]$0.975$[/tex], df = 11 is [tex]$21.92$[/tex]
So, 95% Cl is
[tex]$\left(\frac{v(n-1)\times s}{v21.92}, \frac{v(n-1)\times s}{v3.82}\right)$[/tex]
[tex]$=\frac{\sqrt{11}\times 147.928}{\sqrt{21.92}},\frac{\sqrt{11}\times 147.928}{\sqrt{3.82}}$[/tex]
[tex]$=(104.7916,251.0239)$[/tex]
c). Given a = [tex]$0.01$[/tex], chisquare with [tex]$0.005$[/tex], df = [tex]$11$[/tex] is [tex]$2.6$[/tex]
Chisquare with [tex]$0.995$[/tex], df = [tex]$11$[/tex] is [tex]$26.76$[/tex]
So 99% CI is
[tex]$\left(\frac{v(n-1)\times s}{v26.76}, \frac{v(n-1)\times s}{v2.6}\right)$[/tex]
[tex]$=\frac{\sqrt{11}\times 147.928}{\sqrt{26.76}},\frac{\sqrt{11}\times 147.928}{\sqrt{2.6}}$[/tex]
[tex]$=(94.84265,304.2706)$[/tex]
