| Exam Board | CAIE |
|---|---|
| Module | S1 (Statistics 1) |
| Year | 2016 |
| Session | November |
| Marks | 8 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Normal Distribution |
| Type | Probability calculation plus find unknown boundary |
| Difficulty | Standard +0.3 This is a straightforward normal distribution question requiring standard z-score calculations and inverse normal lookups. All parts follow routine procedures: (i) basic probability calculation, (ii) simple division, (iii) inverse normal to find mean, (iv) repeat of part (ii). No novel problem-solving or conceptual insight 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 |
|---|---|
| M1 | Standardising with no continuity correction, no \(\sqrt{\phantom{x}}\) or squaring |
| Answer | Marks |
|---|---|
| M1 | \(1 - \Phi(\phantom{x})\) (final process) |
| A1 | Or anything in between [3] |
| Answer | Marks |
|---|---|
| B1 | \(\pm 1.76\) to \(1.77\) [1] |
| Answer | Marks |
|---|---|
| B1 | [3] |
| Answer | Marks |
|---|---|
| M1 | Standardising with \(z\)-value, allow \(\sqrt{\phantom{x}}\) or squaring |
| Answer | Marks |
|---|---|
| A1 | Or anything in between, follow-through their (ii) |
| Answer | Marks |
|---|---|
| B1 | [1] |
**(i)**
$P(< 1) = P\left(z < \frac{1-1.04}{0.017}\right) = P(z < -2.353)$
M1 | Standardising with no continuity correction, no $\sqrt{\phantom{x}}$ or squaring
$= 1 - 0.9907 = 0.0093$
M1 | $1 - \Phi(\phantom{x})$ (final process)
A1 | Or anything in between [3]
**(ii)**
B1 | $\pm 1.76$ to $1.77$ [1]
**(iii)**
Expected number $= 1000 \div 1.04 = 961$ or $962$
B1 | [3]
$z = -1.765$
M1 | Standardising with $z$-value, allow $\sqrt{\phantom{x}}$ or squaring
$-1.765 = \frac{1-\mu}{0.017}$
$\mu = 1.03$
A1 | Or anything in between, follow-through their (ii)
Expected number $= 1000 \div 1.03 = 971$ or $970$
B1 | [1]
---4 Packets of rice are filled by a machine and have weights which are normally distributed with mean 1.04 kg and standard deviation 0.017 kg .\\
(i) Find the probability that a randomly chosen packet weighs less than 1 kg .\\
(ii) How many packets of rice, on average, would the machine fill from 1000 kg of rice?
The factory manager wants to produce more packets of rice. He changes the settings on the machine so that the standard deviation is the same but the mean is reduced to $\mu \mathrm { kg }$. With this mean the probability that a packet weighs less than 1 kg is 0.0388 .\\
(iii) Find the value of $\mu$.\\
(iv) How many packets of rice, on average, would the machine now fill from 1000 kg of rice?
\hfill \mbox{\textit{CAIE S1 2016 Q4 [8]}}