21N.2.SL.TZ0.9

pestleMathematicsAASLPaper 221N· sl-4-6-combined-mutually-exclusive-conditional-independence-prob-diagramssource ↗

The random variable $X$ follows a normal distribution with mean $μ$ and standard deviation $σ$ .

The avocados grown on a farm have weights, in grams, that are normally distributed with mean $μ$ and standard deviation $σ$ . Avocados are categorized as small, medium, large or premium, according to their weight. The following table shows the probability an avocado grown on the farm is classified as small, medium, large or premium.

The maximum weight of a small avocado is $106.2$ grams.

The minimum weight of a premium avocado is $182.6$ grams.

A supermarket purchases all the avocados from the farm that weigh more than $106.2$ grams.

Find the probability that an avocado chosen at random from this purchase is categorized as

Find $P (μ - 1.5 σ < X < μ + 1.5 σ)$ .

[3]

Find the value of $μ$ and of $σ$ .

[5]

medium.

[2]

c.i.

large.

[1]

c.ii.

premium.

[1]

c.iii.

The selling prices of the different categories of avocado at this supermarket are shown in the following table:

The supermarket pays the farm $$ 200$ for the avocados and assumes it will then sell them in exactly the same proportion as purchased from the farm.

According to this model, find the minimum number of avocados that must be sold so that the net profit for the supermarket is at least $$ 438$ .

[4]

Markscheme / solution

$P (\frac{μ - 1.5 σ - μ}{σ} < \frac{X - μ}{σ} < \frac{μ + 1.5 σ - μ}{σ})$ (M1)

$P (- 1.5 < Z < 1.5)$ OR $1 - 2 \times P (Z < - 1.5)$ (A1)

$P (- 1.5 < Z < 1.5) = 0.866385 \dots$

$P (μ - 1.5 σ < X < μ + 1.5 σ) = 0.866$ A1

Note: Do not award any marks for use of their answers from part (b).

[3 marks]

$z_{1} = - 1.75068 \dots$ and $z_{2} = 1.30468 \dots$ (seen anywhere) (A1)

correct equations (A1)(A1)

$\frac{106.2 - μ}{σ} = - 1.75068 \dots, μ + 1.30468 \dots σ = 182.6$

attempt to solve their equations involving z values (M1)

$μ = 149.976 . . ., σ = 25.0051 \dots$

$μ = 150, σ = 25.0$ A1

[5 marks]

new sample space is $96 %$ (may be seen in (ii) or (iii)) (M1)

$P (medium | not small)$ OR $\frac{0.576}{0.96}$

$P (Medium) = 0.6$ A1

[2 marks]

c.i.

$P (Large) = 0.3$ A1

[1 mark]

c.ii.

$P (Premium) = 0.1$ A1

[1 mark]

c.iii.

attempt to express revenue from avocados (M1)

$1.1 \times 0.6 + 1.29 \times 0.3 + 1.96 \times 0.1$ OR $1.243 n$

correct inequality or equation for net profit in terms of $n$ (A1)

$1.1 \times 0.6 n + 1.29 \times 0.3 n + 1.96 \times 0.1 n - 200 \geq 438$ OR $1.243 n - 200 = 438$

attempt to solve the inequality (M1)

sketch OR $n = 513.274 . . .$

$n = 514$ A1

Note: Only award follow through in part (d) for 3 probabilities which add up to 1. FT of probabilities from c) that do not add up to 1 should only be awarded M marks, where appropriate, in d).

[4 marks]

Examiners’ report

[N/A]

c.i.

[N/A]

c.ii.

[N/A]

c.iii.

[N/A]