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

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

Write down the number of low production hives.

Find the value of $k$;

Find the number of hives that have a high production.

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.

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]

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

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

1424.67

$P = 1420{\text{ (g)}}$     A1     N2

[2 marks]

40 (hives)     A1     N1

[1 mark]

valid approach     (M1)

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

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

$k = 1640$     A1     N3

[3 marks]

valid approach     (M1)

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

32 (hives)     A1     N2

[2 marks]

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]