Obtain $P(1.65<\overline{X}\leq 4.35,0.12<S^2\leq 55.26)$ If $X_{i}$ follows $N(3,12)$ and $i=1,2,3$.
My approach
Since, $\overline{X}$ and $S^2$ are independent, the given probability can be factored into two individual probabilities. After adjusting terms and making standard Normal and chi square, we have:
$(2\Phi_{0.675}-1)P(0.02<\chi_{2}\leq 9.21)$
Which gives me $(0.9)(0.98)=0.882$.
But none of the options seems to be true. I am not able to identify where the problem is?
Any help?
Thanks