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Date May 2022 Marks available 3 Reference code 22M.1.AHL.TZ1.12
Level Additional Higher Level Paper Paper 1 Time zone Time zone 1
Command term Find Question number 12 Adapted from N/A

Question

The strength of earthquakes is measured on the Richter magnitude scale, with values typically between 0 and 8 where 8 is the most severe.

The Gutenberg–Richter equation gives the average number of earthquakes per year, N, which have a magnitude of at least M. For a particular region the equation is

log10N=a-M, for some a.

This region has an average of 100 earthquakes per year with a magnitude of at least 3.

The equation for this region can also be written as N=b10M.

Within this region the most severe earthquake recorded had a magnitude of 7.2.

The number of earthquakes in a given year with a magnitude of at least 7.2 can be modelled by a Poisson distribution, with mean N. The number of earthquakes in one year is independent of the number of earthquakes in any other year.

Let Y be the number of years between the earthquake of magnitude 7.2 and the next earthquake of at least this magnitude.

Find the value of a.

[2]
a.

Find the value of b.

[2]
b.

Find the average number of earthquakes in a year with a magnitude of at least 7.2.

[1]
c.

Find P(Y>100).

[3]
d.

Markscheme

log10100=a-3        (M1)

a=5             A1

 

[2 marks]

a.

EITHER

N=105-M        (M1)

=10510M=10000010M


OR

100=b103        (M1)


THEN

b=100000  =105             A1

 

[2 marks]

b.

N=105107.2=0.00631   0.0063095           A1


Note: Do not accept an answer of 10-2.2.

 

[1 mark]

c.

METHOD 1

Y>100no earthquakes in the first 100 years             (M1)


EITHER

let X be the number of earthquakes of at least magnitude 7.2 in a year

X~Po0.0063095

PX=0100             (M1)


OR

let X be the number of earthquakes in 100 years

X~Po0.0063095×100             (M1)

PX=0


THEN

0.532  0.532082           A1

 

METHOD 2

Y>100no earthquakes in the first 100 years             (M1)

let X be the number of earthquakes in 100 years

since n is large and p is small

X~B100, 0.0063095             (M1)

PX=0

0.531  0.531019           A1

 

[3 marks]

d.

Examiners report

Parts (a), (b), and (c) were accessible to many candidates who earned full marks with the manipulation of logs and indices presenting no problems. Part (d), however, proved to be too difficult for most and very few correct attempts were seen. As in question 9, most candidates relied on calculator notation when using the Poisson distribution. The discipline of defining a random variable in terms of its distribution and parameters helps to conceptualize the problem in terms that aid a better understanding. Most candidates who attempted this question blindly entered values into the Poisson distribution calculator and were unable to earn any marks. There were a couple of correct solutions using a binomial distribution to approximate the given quantity.

a.
[N/A]
b.
[N/A]
c.
[N/A]
d.

Syllabus sections

Topic 4—Statistics and probability » AHL 4.17—Poisson distribution
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Topic 4—Statistics and probability

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