| Exam Board | OCR |
|---|---|
| Module | H240/02 (Pure Mathematics and Statistics) |
| Year | 2018 |
| Session | March |
| Marks | 9 |
| Topic | Normal Distribution |
| Type | Symmetric properties of normal |
| Difficulty | Standard +0.3 This question requires finding μ and σ from two given conditions (median and 70th percentile), then applying these to calculate further probabilities and inverse normal values. While it involves multiple steps and inverse normal calculations, the techniques are standard for A-level Stats 1 with no novel problem-solving required—slightly above average due to the multi-part nature and need to work backwards initially. |
| Spec | 2.04e Normal distribution: as model N(mu, sigma^2)2.04f Find normal probabilities: Z transformation |
| Answer | Marks | Guidance |
|---|---|---|
| Percentage with masses \(> 59\) g = 30% | B1, B1 | or 0.3 |
| [2] |
| Answer | Marks | Guidance |
|---|---|---|
| \(\Phi\left(\frac{53-56}{\sigma}\right) = 0.3\), \(\frac{53-56}{\sigma} = -0.5244\) | M1 | |
| \(\sigma = 5.721\) | A1 | |
| \(X \sim N(56, 5.721^2)\) soi | M1 | |
| \(\text{P}(X > 65) = 0.0578\) or \(5.78\%\) (3 sf) | A1 | or \(\text{P}(X > 65) = \text{P}(Z > \frac{65-56}{5.721}) = \text{P}(Z > 1.573)\) ft their \(\sigma\) |
| Answer | Marks |
|---|---|
| [4] |
| Answer | Marks |
|---|---|
| \(a = 46.4\) (3 sf) | M1, A1, A1 |
| [3] |
## 8(i)
$n = 56$
Percentage with masses $> 59$ g = 30% | B1, B1 | or 0.3
| [2]
## 8(ii)
$\Phi\left(\frac{53-56}{\sigma}\right) = 0.3$, $\frac{53-56}{\sigma} = -0.5244$ | M1 |
$\sigma = 5.721$ | A1 |
$X \sim N(56, 5.721^2)$ soi | M1 |
$\text{P}(X > 65) = 0.0578$ or $5.78\%$ (3 sf) | A1 | or $\text{P}(X > 65) = \text{P}(Z > \frac{65-56}{5.721}) = \text{P}(Z > 1.573)$ ft their $\sigma$
Or BC
| [4]
## 8(iii)
$\text{P}(X < 50) = 0.1471$
$\text{P}(X < a) = 0.0471)$
$a = 46.4$ (3 sf) | M1, A1, A1 |
| [3]
---8 The masses, $X$ grams, of tomatoes are normally distributed. Half of the tomatoes have masses greater than 56.0 g and $70 \%$ of the tomatoes have masses greater than 53.0 g .\\
(i) Find the percentage of tomatoes with masses greater than 59.0 g .\\
(ii) Find the percentage of tomatoes with masses greater than 65.0 g .\\
(iii) Given that $\mathrm { P } ( a < X < 50 ) = 0.1$, find $a$.
\hfill \mbox{\textit{OCR H240/02 2018 Q8 [9]}}