| Exam Board | CAIE |
|---|---|
| Module | S1 (Statistics 1) |
| Year | 2011 |
| Session | June |
| Marks | 5 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Approximating Binomial to Normal Distribution |
| Type | Justify normal approximation |
| Difficulty | Moderate -0.8 This is a straightforward application of the normal approximation to binomial with n=30, p=0.8. Part (i) requires standard procedure: check conditions, calculate mean/variance, apply continuity correction, and use normal tables. Part (ii) is simple recall of the np>5 and nq>5 conditions. The calculations are routine with no conceptual challenges beyond remembering the continuity correction. |
| Spec | 2.04b Binomial distribution: as model B(n,p)2.04c Calculate binomial probabilities2.04d Normal approximation to binomial |
| Answer | Marks | Guidance |
|---|---|---|
| (i) \(np = 24\), \(npq = 4.8\) | B1 | \(24\) and \(4.8\) or \(\sqrt{4.8}\) seen can be unsimplified |
| \(z = \pm\left(\frac{24.5 - 24}{\sqrt{4.8}}\right) = 0.228\) | M1, M1 | Standardising, need sq rt, cc not necessary; Continuity correction \(24.5\) or \(25.5\) used |
| Prob \(= 0.590\) | A1 [4] | Correct answer must be from \(24.5\) |
| (ii) \(np\) and \(nq\) both \(> 5\) | B1 [1] | Need both |
**(i)** $np = 24$, $npq = 4.8$ | B1 | $24$ and $4.8$ or $\sqrt{4.8}$ seen can be unsimplified
$z = \pm\left(\frac{24.5 - 24}{\sqrt{4.8}}\right) = 0.228$ | M1, M1 | Standardising, need sq rt, cc not necessary; Continuity correction $24.5$ or $25.5$ used
Prob $= 0.590$ | A1 [4] | Correct answer must be from $24.5$
**(ii)** $np$ and $nq$ both $> 5$ | B1 [1] | Need both2 In Scotland, in November, on average $80 \%$ of days are cloudy. Assume that the weather on any one day is independent of the weather on other days.\\
(i) Use a normal approximation to find the probability of there being fewer than 25 cloudy days in Scotland in November (30 days).\\
(ii) Give a reason why the use of a normal approximation is justified.
\hfill \mbox{\textit{CAIE S1 2011 Q2 [5]}}