CAIE A-Level Mathematics 5.5 Normal Distribution Question Bank

Practise direct and inverse normal calculations and normal approximations to binomial counts with continuity correction.

Exam points

standardise a normal variable or use calculator normal CDF for a tail or interval
use inverse normal probabilities to find a cut-off, mean or standard deviation
approximate a binomial with N(np,npq) and apply the correct continuity correction

Covered syllabus points

5.5.1—Normal distribution 5.5.2—Problems concerning a variable X 5.5.3—Binomial distribution

[Maximum number: 7]

The mass of grapes sold per day by a large shop can be modelled by a normal distribution with mean 28 kg . On 10 % of days less than 16 kg of grapes are sold.

(a)

Find the standard deviation of the mass of grapes sold per day.

[ 3 ]

Given P(X<16)=0.1,
$\frac{16-28}{\sigma}=-1.282.$
Hence,
$\sigma=9.36\text{ kg}.$

(b)

In a random sample of 365 days, on how many days would you expect the mass of grapes sold to be within 1.3 standard deviations of the mean?

[ 4 ]

The probability of being within 1.3 standard deviations of the mean is
$P(-1.3<Z<1.3)=2\Phi(1.3)-1=2(0.9032)-1=0.8064.$
In 365 days, the expected number is
365(0.8064)=294.3,
so the answer is 294 days.

[Maximum number: 5]

In the region of Arka, the total number of households in the three villages Reeta, Shan and Teber is 800 . Each of the households was asked about the quality of their broadband service. Their responses are summarised in the following table.

Table for Question 5.5 The normal distribution question 2 — CAIE A-Level Mathematics AS

120 households in Arka are chosen at random.

Use an approximation to find the probability that more than 32 of these households have no broadband service.

\text { Mean }=120 \times 0.35[=42]
Variance $=120 \times 0.35 \times 0.65[=27.3]$
B1
Correct mean and variance seen, allow unsimplified
$\mathrm{P}(X>32)=\mathrm{P}\left(Z>\frac{32.5-42}{\sqrt{27.3}}\right)=\mathrm{P}(Z>-1.818)$
M1
Substituting their mean and variance into
± standardisation formula (any number), condone $\sigma^{2}$ or
$\sqrt{ } \sigma$
M1
Using continuity correction 31.5 or 32.5
$\Phi(1.818)$
M1
Appropriate area $\Phi$ , from final process, must be
probability
0.966
A1
$0.965 \leqslant p \leqslant 0.966$

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Covered syllabus points

5.5 The normal distribution question 1

Question (a)

Question (b)

5.5 The normal distribution question 2