| Exam Board | CAIE |
|---|---|
| Module | S1 (Statistics 1) |
| Year | 2004 |
| Session | June |
| Marks | 8 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Normal Distribution |
| Type | Mixed calculations with boundaries |
| Difficulty | Moderate -0.3 This is a straightforward application of normal distribution with standard z-score calculations. Part (i) requires a single standardization and table lookup, while part (ii) involves finding a percentile (the 54.78th percentile). Both are routine procedures covered early in S1 with no conceptual challenges or multi-step reasoning required. |
| Spec | 2.04e Normal distribution: as model N(mu, sigma^2)2.04f Find normal probabilities: Z transformation |
| Answer | Marks | Guidance |
|---|---|---|
| \(z = \frac{350-450}{120} = -0.833\) | M1 | For standardising; accept 120 or \(\sqrt{120}\), no cc |
| A1 | For correct \(z\) value, \(+\) or \(-\), accept \(\ 0.83\) | |
| \(\%\ \text{small} = 1 - 0.7975 = 0.2025\) or \(20.25\%\) | A1 | For answer rounding to 0.202 or 0.203 |
| Answer | Marks | Guidance |
|---|---|---|
| \(0.7975 \div 2 = 0.39875\) each | M1 | For dividing their remainder by 2 |
| \(\Phi z_2 = 0.60125\) | M1dep | For adding their above two probs together or subtracting from 1 |
| \(z_2 = 0.257\) | M1 | For finding the \(z\) corresponding to their probability |
| \(x = 120 \times 0.257 + 450 = 481\) | M1dep | For converting to \(x\) from a \(z\) value |
| A1 | For answer, rounding to 481 |
# Question 4:
## Part (i)
$z = \frac{350-450}{120} = -0.833$ | M1 | For standardising; accept 120 or $\sqrt{120}$, no cc
| A1 | For correct $z$ value, $+$ or $-$, accept $\ 0.83$
$\%\ \text{small} = 1 - 0.7975 = 0.2025$ or $20.25\%$ | A1 | For answer rounding to 0.202 or 0.203
**Total: 3**
## Part (ii)
$0.7975 \div 2 = 0.39875$ each | M1 | For dividing their remainder by 2
$\Phi z_2 = 0.60125$ | M1dep | For adding their above two probs together or subtracting from 1
$z_2 = 0.257$ | M1 | For finding the $z$ corresponding to their probability
$x = 120 \times 0.257 + 450 = 481$ | M1dep | For converting to $x$ from a $z$ value
| A1 | For answer, rounding to 481
**Total: 5**
---4 Melons are sold in three sizes: small, medium and large. The weights follow a normal distribution with mean 450 grams and standard deviation 120 grams. Melons weighing less than 350 grams are classified as small.\\
(i) Find the proportion of melons which are classified as small.\\
(ii) The rest of the melons are divided in equal proportions between medium and large. Find the weight above which melons are classified as large.
\hfill \mbox{\textit{CAIE S1 2004 Q4 [8]}}