User interface language: English | Español

Date November 2018 Marks available 1 Reference code 18N.2.AHL.TZ0.H_10
Level Additional Higher Level Paper Paper 2 Time zone Time zone 0
Command term Give Question number H_10 Adapted from N/A

Question

Willow finds that she receives approximately 70 emails per working day.

She decides to model the number of emails received per working day using the random variable X , where X follows a Poisson distribution with mean 70.

In order to test her model, Willow records the number of emails she receives per working day over a period of 6 months. The results are shown in the following table.

From the table, calculate

Archie works for a different company and knows that he receives emails according to a Poisson distribution, with a mean of λ emails per day.

Using this distribution model, find  P ( X < 60 ) .

[2]
a.i.

Using this distribution model, find the standard deviation of X .

[2]
a.ii.

an estimate for the mean number of emails received per working day.

[3]
b.i.

an estimate for the standard deviation of the number of emails received per working day.

[2]
b.ii.

Give one piece of evidence that suggests Willow’s Poisson distribution model is not a good fit.

[1]
c.

Suppose that the probability of Archie receiving more than 10 emails in total on any one day is 0.99. Find the value of λ.

[3]
d.

Now suppose that Archie received exactly 20 emails in total in a consecutive two day period. Show that the probability that he received exactly 10 of them on the first day is independent of λ.

[5]
e.

Markscheme

P ( X < 60 )

= P ( X 59 )      (M1)

= 0.102     A1

 

[2 marks]

a.i.

standard deviation =  70 (= 8.37)      (M1)A1

 

[2 marks]

a.ii.

use of midpoints (accept consistent use of 45, 55 etc.)     (M1)

44.5 × 2 + 54.5 × 15 + 64.5 × 40 + 74.5 × 53 + 94.5 + 104.5 × 3 + 114.5 × 6 2 + 15 + 40 + 53 + 0 + 1 + 3 + 6       (M1)

= 8530 120 ( = 71.1 )       A1

Note: If 45, 55, etc. are used consistently instead of midpoints (implied by the answer 71.58…) award M1M1A0.

 

[3 marks]

b.i.

13.9     (M1)A1

 

[2 marks]

b.ii.

valid reason given to include the examples below       R1

variance is 192 which is not close to the mean (accept not equal to) standard deviation too high (using parts (a)(ii) and (b)(ii))

relative frequency of X ≤ 59 is 0.142 which is too high (using part (a)(i))

Poisson would give a frequency of roughly 14 for 80 ≤ X  ≤ 89

Note: Reasons which do not use values found in previous parts must be backed up with numerical evidence.

 

[1 mark]

c.

 

P ( Y > 10 ) = 0.99

1 P ( Y 10 ) = 0.99 P ( Y 10 ) = 0.01       (M1)

attempt to solve a correct equation       (M1)

λ = 20.1       A1

 

[3 marks]

d.

in 1 day, no of emails is X ~ Po(λ)

in 2 days, no of emails is Y ~ Po(2λ)       (A1)

P(10 on first day | 20 in 2 days)        (M1)

P ( X = 10 ) × P ( X = 10 ) P ( Y = 20 )         (M1)

= ( λ 10 e λ 10 ! ) 2 ( 2 λ ) 20 e 2 λ 20 !      A1

= λ 20 e 2 λ 2 20 λ 20 e 2 λ × 20 ! ( 10 ! ) 2       A1

= 20 ! 2 20 ( 10 ! ) 2

which is independent of λ       AG

 

[5 marks]

e.

Examiners report

[N/A]
a.i.
[N/A]
a.ii.
[N/A]
b.i.
[N/A]
b.ii.
[N/A]
c.
[N/A]
d.
[N/A]
e.

Syllabus sections

Topic 4—Statistics and probability » SL 4.1—Concepts, reliability and sampling techniques
Show 80 related questions
Topic 4—Statistics and probability » SL 4.3—Mean, median, mode. Mean of grouped data, standard deviation. Quartiles, IQR
Topic 4—Statistics and probability » SL 4.6—Combined, mutually exclusive, conditional, independence, prob diagrams
Topic 4—Statistics and probability » SL 4.8—Binomial distribution
Topic 4—Statistics and probability » AHL 4.17—Poisson distribution
Topic 4—Statistics and probability

View options