(i)     Determine two simultaneous linear equations satisfied by $\mu$ and $\sigma$.

(ii)     Find the values of $\mu$ and $\sigma$.

Find the probability that an office worker weighs more than $100$ kg.

There are elevators in the office building that take the office workers to their offices.

Given that there are $10$ workers in a particular elevator,

find the probability that at least four of the workers weigh more than $100$ kg.

Given that there are $10$ workers in an elevator and at least one weighs more than $100$ kg,

find the probability that there are fewer than four workers exceeding $100$ kg.

The arrival of the elevators at the ground floor between $08:00$ and $09:00$ can be modelled by a Poisson distribution. Elevators arrive on average every $36$ seconds.

Find the probability that in any half hour period between $08:00$ and $09:00$ more than $60$ elevators arrive at the ground floor.

An elevator can take a maximum of $10$ workers. Given that $400$ workers arrive in a half hour period independently of each other,

find the probability that there are sufficient elevators to take them to their offices.

Note:     In Section B, accept answers that correctly round to 2 sf.

(i)     let $W$ be the weight of a worker and $W \sim {\text{N}}(\mu ,{\text{ }}{\sigma ^2})$

${\text{P}}\left( {Z < \frac{{62 - \mu }}{\alpha }} \right) = 0.3$ and ${\text{P}}\left( {Z < \frac{{98 - \mu }}{\sigma }} \right) = 0.75$     (M1)

Note:     Award M1 for a correctly shaded and labelled diagram.

$\frac{{62 - \mu }}{\sigma } = {\Phi ^{ - 1}}(0.3)\;\;\;( =  - 0.524 \ldots )\;\;\;$and

$\frac{{98 - \mu }}{\sigma } = {\Phi ^{ - 1}}(0.75)\;\;\;( = 0.674 \ldots )$

or linear equivalents     A1A1

Note:     Condone equations containing the GDC inverse normal command.

(ii)     attempting to solve simultaneously     (M1)

$\mu  = 77.7,{\text{ }}\sigma  = 30.0$     A1A1

[6 marks]

Note:     In Section B, accept answers that correctly round to 2 sf.

${\text{P}}(W > 100) = 0.229$     A1

[1 mark]

Note:     In Section B, accept answers that correctly round to 2 sf.

let $X$ represent the number of workers over $100$ kg in a lift of ten passengers

$X \sim {\text{B}}(10,{\text{ }}0.229 \ldots )$     (M1)

${\text{P}}(X \ge 4) = 0.178$     A1

[2 marks]

Note:     In Section B, accept answers that correctly round to 2 sf.

${\text{P}}(X < 4|X \ge 1) = \frac{{{\text{P}}(1 \le X \le 3)}}{{{\text{P}}(X \ge 1)}}$     M1(A1)

Note:     Award the M1 for a clear indication of a conditional probability.

$= 0.808$     A1

[3 marks]

Note:     In Section B, accept answers that correctly round to 2 sf.

$L \sim {\text{Po}}(50)$     (M1)

${\text{P}}(L > 60) = 1 - {\text{P}}(L \le 60)$     (M1)

$= 0.0722$     A1

[3 marks]

Note:     In Section B, accept answers that correctly round to 2 sf.

$400$ workers require at least $40$ elevators     (A1)

${\text{P}}(L \ge 40) = 1 - {\text{P}}(L \le 39)$     (M1)

$= 0.935$     A1

[3 marks]

Total [18 marks]