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

a.(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]