Solved: The weights, in grams, of individual packets of coffee can be modelled by a... [IB Mathematics Applications & Interpretation (Math AI)]

Question

HLPaper 2

The weights, in grams, of individual packets of coffee can be modelled by a normal distribution,with mean $102 g$ and standard deviation $8 g$ .

Find the probability that a randomly selected packet has a weight less than $100 g$ .

[2]

Verified

Solution

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

$Χ ~ N (102, 8^{2})$

$P (Χ < 100) = 0.401$ (M1)A1

[2 marks]

The probability that a randomly selected packet has a weight greater than $w$ gramsis $0.444$ . Find the value of $w$ .

[2]

Verified

Solution

$P (Χ > w) = 0.444$ (M1)

$\Rightarrow w = 103 (g)$ A1

[2 marks]

A packet is randomly selected. Given that the packet has a weight greater than $105 g$ ,find the probability that it has a weight greater than $110 g$ .

[3]

Verified

Solution

$P (Χ > 100 Χ > 105) = \frac{P (Χ > 100 \cap Χ > 105)}{P (Χ > 105)}$ (M1)

$= \frac{P (Χ > 100)}{P (Χ > 105)}$ (A1)

$= \frac{0.15865 \dots}{0.35383 \dots}$

$= 0.448$ A1

[3 marks]

From a random sample of $500$ packets, determine the number of packets that would beexpected to have a weight lying within $1.5$ standard deviations of the mean.

[3]

Verified

Solution

EITHER

$P (90 < Χ < 114) = 0.866 \dots$ (A1)

$P (- 1.5 < Z < 1.5) = 0.866 \dots$ (A1)

THEN

$0.866 \dots \times 500$ (M1)

$= 433$ A1

[3 marks]

Packets are delivered to supermarkets in batches of $80$ . Determine the probability thatat least $20$ packets from a randomly selected batch have a weight less than $95 g$ .

[4]

Verified

Solution

$p = P (Χ < 95) = 0.19078 \dots$ (A1)

recognising $Y ~ B (80, p)$ (M1)

now using $Y ~ B (80, 0.19078 \dots)$ (M1)

$P (Y \geq 20) = 0.116$ A1

[4 marks]

The weights, in grams, of individual packets of coffee can be modelled by a normal distribution,with mean 102 g and standard deviation 8 g.

Related topics