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Date	November 2017	Marks available	3	Reference code	17N.2.SL.TZ0.S_8
Level	Standard Level	Paper	Paper 2	Time zone	Time zone 0
Command term	Find	Question number	S_8	Adapted from	N/A

Question

Adam is a beekeeper who collected data about monthly honey production in his bee hives. The data for six of his hives is shown in the following table.

N17/5/MATME/SP2/ENG/TZ0/08

The relationship between the variables is modelled by the regression line with equation $P = aN + b$ .

Adam has 200 hives in total. He collects data on the monthly honey production of all the hives. This data is shown in the following cumulative frequency graph.

N17/5/MATME/SP2/ENG/TZ0/08.c.d.e

Adam’s hives are labelled as low, regular or high production, as defined in the following table.

N17/5/MATME/SP2/ENG/TZ0/08.c.d.e_02

Adam knows that 128 of his hives have a regular production.

Write down the value of $a$ and of $b$ .

[3]

a.

Use this regression line to estimate the monthly honey production from a hive that has 270 bees.

[2]

b.

Write down the number of low production hives.

[1]

c.

Find the value of $k$ ;

[3]

d.i.

Find the number of hives that have a high production.

[2]

d.ii.

Adam decides to increase the number of bees in each low production hive. Research suggests that there is a probability of 0.75 that a low production hive becomes a regular production hive. Calculate the probability that 30 low production hives become regular production hives.

[3]

e.

Markscheme

* This question is from an exam for a previous syllabus, and may contain minor differences in marking or structure.

evidence of setup (M1)

eg $\,\,\,\,\,$ correct value for $a$ or $b$

$a = 6.96103,{\text{ }}b = - 454.805$

$a = 6.96,{\text{ }}b = - 455{\text{ (accept }}6.96x - 455)$ A1A1 N3

[3 marks]

a.

substituting $N = 270$ into their equation (M1)

eg $\,\,\,\,\,$ $6.96(270) - 455$

1424.67

$P = 1420{\text{ (g)}}$ A1 N2

[2 marks]

b.

40 (hives) A1 N1

[1 mark]

c.

valid approach (M1)

eg $\,\,\,\,\,$ $128 + 40$

168 hives have a production less than $k$ (A1)

$k = 1640$ A1 N3

[3 marks]

d.i.

valid approach (M1)

eg $\,\,\,\,\,$ $200 - 168$

32 (hives) A1 N2

[2 marks]

d.ii.

recognize binomial distribution (seen anywhere) (M1)

eg $\,\,\,\,\,$ $X \sim {\text{B}}(n,{\text{ }}p),{\text{ }}\left( {\begin{array}{*{20}{c}} n \\ r \end{array}} \right){p^r}{(1 - p)^{n - r}}$

correct values (A1)

eg $\,\,\,\,\,$ $n = 40$ (check FT) and $p = 0.75$ and $r = 30,{\text{ }}\left( {\begin{array}{*{20}{c}} {40} \\ {30} \end{array}} \right){0.75^{30}}{(1 - 0.75)^{10}}$

0.144364

0.144 A1 N2

[3 marks]

e.

Examiners report

[N/A]

a.

[N/A]

b.

[N/A]

c.

[N/A]

d.i.

[N/A]

d.ii.

[N/A]

e.

Syllabus sections

Topic 4—Statistics and probability » SL 4.2—Histograms, CF graphs, box plots

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