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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 $σ$ .

[3]

Find the probability that a randomly selected flight will have a flight time of more than $80$ minutes.

[2]

Given that a flight between the two cities takes longer than $80$ minutes, find the probability that it takes less than $82$ minutes.

[4]

Find the expected number of flights that will have a flight time of more than $80$ minutes.

[3]

Find the probability that more than $6$ of the flights on this particular day will have a flight time of more than $80$ minutes.

[3]

Markscheme

use of inverse normal to find $z$ -score (M1)

$z = 2.0537 \dots$

$2.0537 \dots = \frac{82 - 75}{σ}$ (A1)

$σ = 3.408401 \dots$

$σ = 3.41$ A1

[3 marks]

evidence of identifying the correct area under the normal curve (M1)

$P (T > 80) = 0.071193 \dots$

$P (T > 80) = 0.0712$ A1

[2 marks]

recognition that $P (80 < T < 82)$ is required (M1)

$P (T < 82 T > 80) = \frac{P (80 < T < 82)}{P (T > 80)} = (\frac{0.051193 \dots}{0.071193 \dots})$ (M1)(A1)

$= 0.719075 \dots$

$= 0.719$ A1

[4 marks]

recognition of binomial probability (M1)

$X ~ B (64, 0.071193 \dots)$ or $E (X) = 64 \times 0.071193 \dots$ (A1)

$E (X) = 4.556353 \dots$

$E (X) = 4.56$ (flights) A1

[3 marks]

$P (X > 6) = P (X \geq 7) = 1 - P (X \leq 6)$ (M1)

$= 1 - 0.83088 \dots$ (A1)

$= 0.1691196 \dots$

$= 0.169$ A1

[3 marks]

Examiners report

[N/A]

Syllabus sections

Topic 4—Statistics and probability » SL 4.8—Binomial distribution

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Hide related questions

Topic 4—Statistics and probability » SL 4.9—Normal distribution and calculations

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Topic 4—Statistics and probability

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