CAIE A-Level Mathematics 5.5.3 Normal Approximation to the Binomial

Practise checking whether a binomial variable can be approximated normally, using mean np and variance npq and applying continuity corrections to integer boundaries.

Exam points

verify that both np and nq exceed 5 before replacing B(n, p) by N(np, npq)
convert strict or inclusive integer bounds to half-unit boundaries for continuity correction
standardise each corrected bound with mean np and deviation √(npq), then combine Φ areas

[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 5.5.3—Binomial distribution question 1 — CAIE A-Level Mathematics A2

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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5.5.3—Binomial distribution question 1