| Exam Board | CAIE |
|---|---|
| Module | S1 (Statistics 1) |
| Year | 2014 |
| Session | June |
| Marks | 6 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Approximating Binomial to Normal Distribution |
| Type | Justify normal approximation |
| Difficulty | Moderate -0.3 This is a straightforward application of the normal approximation to the binomial distribution with standard steps: calculate mean and variance, apply continuity correction, and standardize. Part (ii) requires only recall of the standard justification (np and nq both >5). The calculations are routine with no conceptual challenges beyond A-level expectations. |
| Spec | 2.04c Calculate binomial probabilities2.04d Normal approximation to binomial |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| \(np = 252 \times \frac{1}{7} = 36\), \(npq = 252 \times \frac{1}{7} \times \frac{6}{7} = 30.857\) | B1 | Unsimplified 36 and 30.857 seen, oe |
| \(P\!\left(z < \left(\frac{29.5-36}{\sqrt{30.857}}\right)\right) + P\!\left(z > \left(\frac{44.5-36}{\sqrt{30.857}}\right)\right)\) | M1 | Any standardising, sq rt needed |
| M1 | Any continuity correction either 29.5, 30.5, 43.5, 44.5 | |
| \(= P(z < -1.170) + P(z > 1.530)\) | ||
| \(= 1 - 0.8790 + 1 - 0.9370\) | M1 | Correct area \(2 - (\Phi_1 + \Phi_2)\) |
| \(= 0.184\) | A1 | Correct answer — Total: 5 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| \(np\) and \(nq\) are both \(> 5\) | B1 | Must have both — Total: 1 |
## Question 2:
### Part (i)
| Answer/Working | Mark | Guidance |
|---|---|---|
| $np = 252 \times \frac{1}{7} = 36$, $npq = 252 \times \frac{1}{7} \times \frac{6}{7} = 30.857$ | B1 | Unsimplified 36 and 30.857 seen, oe |
| $P\!\left(z < \left(\frac{29.5-36}{\sqrt{30.857}}\right)\right) + P\!\left(z > \left(\frac{44.5-36}{\sqrt{30.857}}\right)\right)$ | M1 | Any standardising, sq rt needed |
| | M1 | Any continuity correction either 29.5, 30.5, 43.5, 44.5 |
| $= P(z < -1.170) + P(z > 1.530)$ | | |
| $= 1 - 0.8790 + 1 - 0.9370$ | M1 | Correct area $2 - (\Phi_1 + \Phi_2)$ |
| $= 0.184$ | A1 | Correct answer — **Total: 5** |
### Part (ii)
| Answer/Working | Mark | Guidance |
|---|---|---|
| $np$ and $nq$ are both $> 5$ | B1 | Must have both — **Total: 1** |
---2 There is a probability of $\frac { 1 } { 7 }$ that Wenjie goes out with her friends on any particular day. 252 days are chosen at random.\\
(i) Use a normal approximation to find the probability that the number of days on which Wenjie goes out with her friends is less than than 30 or more than 44.\\
(ii) Give a reason why the use of a normal approximation is justified.
\hfill \mbox{\textit{CAIE S1 2014 Q2 [6]}}