(i) As part of a recruitment exercise candidates are required to complete three separate tasks. The times taken, $A$, $B$ and $C$, in minutes, for candidates to complete the three tasks are such that

$A \sim N(21, 2^2)$, $B \sim N(32, 7^2)$ and $C \sim N(45, 9^2)$

The time taken by an individual candidate to complete each task is assumed to be independent of the time taken to complete each of the other tasks.

A candidate is selected at random.

(a) Find the probability that the candidate takes a total time of more than 90 minutes to complete all three tasks.

(b) Find $P(A > B)$

(ii) A simple random sample, $X_1, X_2, X_3, X_4$, is taken from a normal population with mean $\mu$ and standard deviation $\sigma$

Given that

$\overline{X} = \frac{X_1 + X_2 + X_3 + X_4}{4}$

and that

$P(X_1 > \overline{X} + k\sigma) = 0.1$

where $k$ is a constant,

find the value of $k$, giving your answer correct to 3 significant figures. You should show all stages of your working.