7.(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

$$A \sim \mathrm {~N} \left( 21,2 ^ { 2 } \right) , B \sim \mathrm {~N} \left( 32,7 ^ { 2 } \right) \text { and } C \sim \mathrm {~N} \left( 45,9 ^ { 2 } \right)$$

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.
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\item Find the probability that the candidate takes a total time of more than 90 minutes to complete all three tasks.
\item Find $\mathrm { P } ( A > B )$\\
(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

$$\bar { X } = \frac { X _ { 1 } + X _ { 2 } + X _ { 3 } + X _ { 4 } } { 4 }$$

and that

$$\mathrm { P } \left( X _ { 1 } > \bar { X } + k \sigma \right) = 0.1$$

where $k$ is a constant,\\
find the value of $k$ ,giving your answer correct to 3 significant figures.

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\hfill \mbox{\textit{Edexcel S3 2018 Q7 [15]}}