CAIE A-Level Mathematics AS 5.5.3 Normal Approximation to Binomial

[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 CAIE A-Level Mathematics AS 5.5.3 Binomial Distribution Questions question 1 — 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 $=120 \times 0.35 \times 0.65[=27.3]$
Correct mean and variance seen, allow unsimplified
$\mathrm{P}(X>32)=\mathrm{P}\left(Z>\frac{32.5-42}{\sqrt{27.3}}\right)=\mathrm{P}(Z>-1.818)$
Substituting their mean and variance into
± standardisation formula (any number), condone $\sigma^{2}$ or
$\sqrt{ } \sigma$
Using continuity correction 31.5 or 32.5
$\Phi(1.818)$
Appropriate area $\Phi$ , from final process, must be
probability
0.966
$0.965 \leqslant p \leqslant 0.966$

CAIE A-Level Mathematics AS 5.5.3 Binomial Distribution Questions question 1