| Exam Board | CAIE |
|---|---|
| Module | S1 (Statistics 1) |
| Year | 2015 |
| Session | November |
| Marks | 7 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Normal Distribution |
| Type | Expected frequency with unknown parameter |
| Difficulty | Standard +0.3 This is a straightforward two-part normal distribution question requiring inverse normal calculation to find σ from a given percentage, then using that σ to find probabilities and expected frequencies. Both parts use standard techniques taught in S1 with no conceptual challenges beyond routine application of normal distribution tables and z-score formulas. |
| Spec | 2.04e Normal distribution: as model N(mu, sigma^2)2.04f Find normal probabilities: Z transformation |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Marks | Guidance |
| \(z = 1.127\) | B1 | \(\pm 1.127\) seen, accept rounding to \(\pm 1.13\) |
| \(1.127 = \frac{136-125}{\sigma}\) | M1 | Standardising, no cc, no sq rt, with attempt at \(z\) |
| \(\sigma = 9.76\) | A1 (3) | (not \(\pm 0.8078, \pm 0.5517, \pm 0.13, \pm 0.87\)). Correct answer |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Marks | Guidance |
\(P(131| M1 |
Standardising once with their sd, no \(\sqrt{}\), \({}^2\), allow cc |
|
| \(= \Phi(1.639) - \Phi(0.6147)\) | M1 | Correct area \(\Phi 2 - \Phi 1\) |
| \(= 0.9493 - 0.7307 = 0.2186\) | M1 | Mult by 170, \(P<1\) |
| Number \(= 0.2186\times 170 = 37\) or 38 or awrt 37.2 | A1 (4) | Correct answer, nfww |
## Question 4:
### Part (i):
| Answer/Working | Marks | Guidance |
|---|---|---|
| $z = 1.127$ | B1 | $\pm 1.127$ seen, accept rounding to $\pm 1.13$ |
| $1.127 = \frac{136-125}{\sigma}$ | M1 | Standardising, no cc, no sq rt, with attempt at $z$ |
| $\sigma = 9.76$ | A1 (3) | (not $\pm 0.8078, \pm 0.5517, \pm 0.13, \pm 0.87$). Correct answer |
### Part (ii):
| Answer/Working | Marks | Guidance |
|---|---|---|
| $P(131<x<141) = P\!\left(\frac{131-125}{9.76}<z<\frac{141-125}{9.76}\right)$ | M1 | Standardising once with their sd, no $\sqrt{}$, ${}^2$, allow cc |
| $= \Phi(1.639) - \Phi(0.6147)$ | M1 | Correct area $\Phi 2 - \Phi 1$ |
| $= 0.9493 - 0.7307 = 0.2186$ | M1 | Mult by 170, $P<1$ |
| Number $= 0.2186\times 170 = 37$ or 38 or awrt 37.2 | A1 (4) | Correct answer, nfww |
---4 The time taken for cucumber seeds to germinate under certain conditions has a normal distribution with mean 125 hours and standard deviation $\sigma$ hours.\\
(i) It is found that $13 \%$ of seeds take longer than 136 hours to germinate. Find the value of $\sigma$.\\
(ii) 170 seeds are sown. Find the expected number of seeds which take between 131 and 141 hours to germinate.
\hfill \mbox{\textit{CAIE S1 2015 Q4 [7]}}