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Date November 2017 Marks available 3 Reference code 17N.2.sl.TZ0.8
Level SL only Paper 2 Time zone TZ0
Command term Calculate Question number 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

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 5 - Statistics and probability » 5.8 » Binomial distribution.
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