The concentration of xenon, $C$ ppm, in a specific gas mixture is modeled by a normal distribution with a mean of 45.2 and a standard deviation of 1.2.

(a) (i) Find the probability that a randomly selected gas sample has a xenon concentration of exactly 45.2 ppm.

(a) (ii) Find the probability that a randomly selected gas sample has a xenon concentration less than 43.1 ppm.

The thickness of a protective gold coating, $T$ $\mu$m, on a circuit component is modeled by a normal distribution with mean $\mu$ and standard deviation $\sigma$.

(b) (i) Given that $P(T<15.5)=0.95$, show that

$$15.5-\mu =1.645\sigma$$

where the constant is given to three decimal places. Fully justify your answer.

(b) (ii) Given further that $P(T<12.0)=0.20$, find the values of $\mu$ and $\sigma$.