Edexcel IAL Maths S3.1.1 Normal Combinations

[Maximum number: 7]

At a particular supermarket, the times taken to serve each customer in a queue at a standard checkout may be modelled by a normal distribution with mean 240 seconds and standard deviation 20 seconds.

There is a queue of 3 customers at a standard checkout.
Making a reasonable assumption about the times taken to serve these customers,

(a)

find the probability that the total time taken to serve the 3 customers will be less than 11 minutes.

[ 3 ]

Let P= time to serve a customer at a standard checkout.
$Q=P_1+P_2+P_3,\quad Q\sim N(720,1200)$ $P(Q<660)=P\left(Z<\frac{660-720}{\sqrt{1200}}\right)=P(Z<-1.732\ldots)$ =0.0418
Calculator gives 0.04163...
awrt 0.041 / 0.042
ALT
B1 for $Q\sim N(12,\frac13)$ .
M1 for $P(Q<11)=P\left(Z<\frac{11-12}{\sqrt{1/3}}\right)$ .

(b)

find the probability that the total time taken to serve the 3 customers at the standard checkout will exceed the total time taken to serve the 7 customers at the express checkout.

[ 4 ]

Let R= time to serve a customer at an express checkout.
$S=(P_1+P_2+P_3)-(R_1+\cdots+R_7),\quad S\sim N(20,1648)$ $P(S>0)=P\left(Z>\frac{0-20}{\sqrt{1648}}\right)=P(Z>-0.492\ldots)$ =0.6879
Calculator gives 0.6888...
awrt 0.688 / 0.689

Edexcel A-Level Mathematics A2 S3.1.1 Linear Combinations of Normal Variables Questions

Exam points

Edexcel A-Level Mathematics A2 S3.1.1 Linear Combinations of Normal Variables Questions question 1

Question (a)

Question (b)