June 2024 Paper 2 Q9
9
Find \(\mathrm{P}(1000 \lt M \lt 1005)\). [1]
You are given that 20% of bags have masses greater than 502 g and 30% of bags have masses less than 499 g.
Determine the values of \(\mu\) and \(\sigma\). Give your answers correct to 2 decimal places. [5]
| Scheme | Marks | AO |
|---|---|---|
| 0.886 (3 sf) | B1 | 1.1 |
| [1] |
Notes
B1: awrt 0.886 (Condone 88.6%)
| Scheme | Marks | AO |
|---|---|---|
| \(\Phi(0.8)\) and \(\Phi(0.3)\) | M1* | 3.1a |
| \(0.84162 = \frac{502 - \mu}{\sigma}\) and \(-0.52440 = \frac{499 - \mu}{\sigma}\) | A1 | 3.4 |
| \(\dfrac{0.84162}{-0.52440} = \dfrac{502 - \mu}{499 - \mu}\) oe or \((0.84162 + 0.52440)\sigma = 502 - 499\) oe | M1 dep* | 1.1 |
| \(\mu = 500.15\) (2 dp) and \(\sigma = 2.20\) (2 dp) | A1 A1 | 1.1 1.1 |
| [5] |
Notes
M1*: Attempted, values not required and allow truncated or rounded values e.g. 0.84… etc. throughout.
May be implied by \(\pm 0.84162\) & \(\pm 0.52440\)
A1: Allow \(\Phi(0.8)\) and \(\Phi(0.3)\) – or use of these expressions throughout (need not see values for \(\Phi\) but if given they must be correct)
M1 dep*: One correct equation in one unknown (FT their \(\Phi\) values) (may be implied by correct answers).
A1: Correct answers to 2sf (500 and 2.2) or better
A1: Correct answers to exactly 2dp, cao
SCB1 for one correct answer to 2dp (max 4/5) or if one/both answers are given with no/insufficient working (max 1/5).
| Scheme | Marks | AO |
|---|---|---|
| (i) 0.5 and 3.5 | B1 | 1.1 |
| [1] | ||
| (ii) 1.5 Inflection points are one sd from mean | B1FT | 1.2 |
| [1] |
Notes
(c)(i) B1: Allow 0 to 1, 3 to 4
(c)(ii) B1FT: \(\dfrac{\text{Their ‘}3.5\text{’} - \text{‘}0.5\text{’}}{2}\) or \(|2 - \text{‘}3.5\text{’}|\) or \(|2 - \text{‘}0.5\text{’}|\)
FT their points of inflection (Note that \(\mu \approx 2\))
Need value and explanation (calculation alone is not sufficient)