S2 June 2017 Q5
5. The time taken for a randomly selected person to complete a test is \(M\) minutes, where \(M \sim \mathrm{N}(14, \sigma^2)\)
Given that 10% of people take less than 12 minutes to complete the test,
Graham selects 15 people at random.
Jovanna takes a random sample of \(n\) people.
Using a normal approximation, the probability that fewer than 9 of these \(n\) people will take less than 12 minutes to complete the test is 0.3085 to 4 decimal places.
| Scheme | Marks |
|---|---|
| \(\mathrm{P}(M \lt 12) = \mathrm{P}\left(Z \lt \dfrac{12 - 14}{\sigma}\right) = 0.1\) | |
| \(\Rightarrow \dfrac{12 - 14}{\sigma} =, \ -1.2816\) | M1 B1 |
| \(\sigma = 1.5605\ldots\) = awrt 1.56 minutes | A1 |
| (3) |
Notes
M1 standardising (\(\pm\)) with 12, 14 and \(\sigma\) and setting equal to a \(z\) value where \(|z| \gt 1\)
B1 \(\pm\)1.2816 or better
A1 awrt 1.56 Do not allow answer written as an exact fraction.
(corrected from the printed mark scheme: the first line prints \(\mathrm{P}(M \lt 10)\); the 12 in the standardisation shows it should be \(\mathrm{P}(M \lt 12)\))
| Scheme | Marks |
|---|---|
| \(T\) represents number less than 12 minutes. \(T \sim \mathrm{B}(15, 0.1)\) | B1 |
| \(\mathrm{P}(T \leqslant 1)\) | M1 |
| \(= 0.549\) | A1 |
| (3) |
Notes
B1 Writing or using B(15, 0.1).
M1 writing \(\mathrm{P}(T \leqslant 1)\) or \(\mathrm{P}(T \lt 2)\) any letter may be used.
A1 awrt 0.549
NB 0.549 gets B1 M1 A1
| Scheme | Marks |
|---|---|
| [\(T\) ~ number of people who take less than 12 mins to complete the test] \(T \sim \mathrm{B}(n, 0.1)\) | |
| \(T\) can be approximated by \(\mathrm{N}(0.1n, 0.09n)\) | B1 |
| \(\mathrm{P}\left(Z \lt \dfrac{8.5 - 0.1n}{\sqrt{0.09n}}\right) = 0.3085\) | M1 |
| \(\dfrac{8.5 - 0.1n}{\sqrt{0.09n}} = -0.5\) or \(\dfrac{8.5 - 0.1x^2}{0.3x} = -0.5\) | B1 M1 A1 |
| \(0.1n - 0.15\sqrt{n} - 8.5 = 0\) \(\sqrt{n} = 10\) | M1A1 |
| \(n = 100\) | A1cso |
| (8) | |
| (14 marks) |
Notes
B1 mean = \(0.1n\) and Var = \(0.09n\) oe may be seen in an attempt at standardisation
M1 using a continuity correction either 8.5 or 7.5 in an attempt at standardised form. Allow 0.09 for sd.
B1 a \(z\) value of awrt \(\pm\) 0.5
M1 standardising using their mean and sd. (If these have not been given then they must be correct here) and one of 7.5, 8, 8.5, 9 or 9.5 and equal to a \(z\) value where \(|z| \gt 0.4\). Allow any form
A1 A correct equation in any form. ISW. Do not allow if they have \(0.3n\) rather than \(0.3\sqrt{n}\)
M1 using either the quadratic formula or completing the square or factorising or any correct method to solve their 3 term quadratic. If they write the quadratic formula down then allow one slip. If no formula written down then it must be correct for their equation. May be implied by seeing 10 or 8.5. They must show working if the equation used is not correct.
2nd A1 awrt 10.0 – do not need to see \(n\) or \(\sqrt{n}\). Allow \(n = 10\) May be implied by 100
3rd A1 cso 100 If they have a second answer of 72.25 they must reject it to get this final mark.