(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]