Date | May 2021 | Marks available | 3 | Reference code | 21M.2.SL.TZ2.8 |
Level | Standard Level | Paper | Paper 2 | Time zone | Time zone 2 |
Command term | Find | Question number | 8 | Adapted from | N/A |
Question
The flight times, T minutes, between two cities can be modelled by a normal distribution with a mean of 75 minutes and a standard deviation of σ minutes.
On a particular day, there are 64 flights scheduled between these two cities.
Given that 2% of the flight times are longer than 82 minutes, find the value of σ.
Find the probability that a randomly selected flight will have a flight time of more than 80 minutes.
Given that a flight between the two cities takes longer than 80 minutes, find the probability that it takes less than 82 minutes.
Find the expected number of flights that will have a flight time of more than 80 minutes.
Find the probability that more than 6 of the flights on this particular day will have a flight time of more than 80 minutes.
Markscheme
use of inverse normal to find z-score (M1)
z=2.0537…
2.0537…=82-75σ (A1)
σ=3.408401…
σ=3.41 A1
[3 marks]
evidence of identifying the correct area under the normal curve (M1)
P(T>80)=0.071193…
P(T>80)=0.0712 A1
[2 marks]
recognition that P(80<T<82) is required (M1)
P(T<82 T>80)=P(80<T<82)P(T>80)=(0.051193…0.071193…) (M1)(A1)
=0.719075…
=0.719 A1
[4 marks]
recognition of binomial probability (M1)
X~B(64, 0.071193…) or E(X)=64×0.071193… (A1)
E(X)=4.556353…
E(X)=4.56 (flights) A1
[3 marks]
P(X>6)=P(X≥7)=1-P(X≤6) (M1)
=1-0.83088… (A1)
=0.1691196…
=0.169 A1
[3 marks]