| Exam Board | CAIE |
|---|---|
| Module | S1 (Statistics 1) |
| Year | 2014 |
| Session | November |
| Marks | 10 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Normal Distribution |
| Type | Symmetric properties of normal |
| Difficulty | Moderate -0.3 This is a straightforward application of normal distribution properties requiring standard z-score calculations and use of symmetry. Part (i) is routine probability calculation with tables, part (ii) tests understanding of symmetry (trivial once recognized), and part (iii) is inverse normal lookup. All are standard textbook exercises with no novel problem-solving required, making it slightly easier than average. |
| Spec | 2.04e Normal distribution: as model N(mu, sigma^2)2.04f Find normal probabilities: Z transformation |
| Answer | Marks | Guidance |
|---|---|---|
| (i) \(z_1 = \frac{70 - 66.4}{5.6} = 0.6429\); \(z_2 = \frac{72.5 - 66.4}{5.6} = 1.089\) | M1 M1 | Standardising one variable, no cc, no sq rt; Correct area \(\Phi_2 - \Phi_1\) |
| \(\Phi(1.089) - \Phi(0.643) = 0.8620 - 0.7399 = 0.1221\); \(0.1221 \times 250 = 30.5\); 30 or 31 sheep | A1 M1 A1 | Correct answer rounding to 0.12; Mult by 250; Correct answer ft their 0.1221 |
| 5 marks | ||
| (ii) \(66.4 - 59.2 = 7.2\); \(66.4 + 7.2 = 73.6\) | M1 A1 | Subt from 66.4; Correct answer |
| 2 marks | ||
| (iii) \(z = 0.674\); \(\frac{67.5 - \mu}{4.92} = 0.674\); \(\mu = 64.2\) | B1 M1 A1 | \(\pm 0.674\) or 0.675 seen; Standardising with a z-value no cc no sq rt; Correct answer |
| 3 marks |
**(i)** $z_1 = \frac{70 - 66.4}{5.6} = 0.6429$; $z_2 = \frac{72.5 - 66.4}{5.6} = 1.089$ | M1 M1 | Standardising one variable, no cc, no sq rt; Correct area $\Phi_2 - \Phi_1$ |
| $\Phi(1.089) - \Phi(0.643) = 0.8620 - 0.7399 = 0.1221$; $0.1221 \times 250 = 30.5$; 30 or 31 sheep | A1 M1 A1 | Correct answer rounding to 0.12; Mult by 250; Correct answer ft their 0.1221 |
| | | 5 marks |
**(ii)** $66.4 - 59.2 = 7.2$; $66.4 + 7.2 = 73.6$ | M1 A1 | Subt from 66.4; Correct answer |
| | | 2 marks |
**(iii)** $z = 0.674$; $\frac{67.5 - \mu}{4.92} = 0.674$; $\mu = 64.2$ | B1 M1 A1 | $\pm 0.674$ or 0.675 seen; Standardising with a z-value no cc no sq rt; Correct answer |
| | | 3 marks |
---6 A farmer finds that the weights of sheep on his farm have a normal distribution with mean 66.4 kg and standard deviation 5.6 kg .\\
(i) 250 sheep are chosen at random. Estimate the number of sheep which have a weight of between 70 kg and 72.5 kg .\\
(ii) The proportion of sheep weighing less than 59.2 kg is equal to the proportion weighing more than $y \mathrm {~kg}$. Find the value of $y$.
Another farmer finds that the weights of sheep on his farm have a normal distribution with mean $\mu \mathrm { kg }$ and standard deviation 4.92 kg . 25\% of these sheep weigh more than 67.5 kg .\\
(iii) Find the value of $\mu$.
\hfill \mbox{\textit{CAIE S1 2014 Q6 [10]}}