June 2022 Paper 3 Q14
14 A customer service centre records every call they receive.
It is found that 30% of all calls made to this centre are complaints.
A sample of 20 calls is selected.
The number of calls in the sample which are complaints is denoted by the random variable \(X\).
(a) State two assumptions necessary for \(X\) to be modelled by a binomial distribution. [2 marks]
(b) Assume that \(X\) can be modelled by a binomial distribution.
(i) Find \(\mathrm{P}(X = 1)\) [1 mark]
(ii) Find \(\mathrm{P}(X \lt 4)\) [2 marks]
(iii) Find \(\mathrm{P}(X \geqslant 10)\) [2 marks]
(c) In a random sample of 10 calls to a school, the number of calls which are complaints, \(Y\), may be modelled by a binomial distribution\[Y \sim \mathrm{B}(10, p)\]
The standard deviation of \(Y\) is 1.5
Calculate the possible values of \(p\). [3 marks]
| Scheme | Marks | AO |
|---|---|---|
States one of the following assumptions in context
Do not allow probability being independent Do not allow fixed number of calls | B1 | 3.5b |
| States a second assumption in context | B1 | 3.5b |
| (2) |
Typical solution
The probability of getting a complaint call is fixed
Calls occur independently of each other
| Scheme | Marks | AO |
|---|---|---|
| (i) Calculates the correct probability ACF AWFW [0.0068, 0.007] | B1 | 1.1b |
| (1) | ||
| (ii) Finds \(P(X \leqslant 4)\) or \(P(X \leqslant 3)\) PI by 0.2375 or 0.107 Allow figures to 2sf | M1 | 1.1a |
| Obtains correct probability ACF AWFW [0.107, 0.11] | A1 | 1.1b |
| (2) | ||
| (iii) Finds their \(P(X \leqslant 9)\) or \(P(X \leqslant 10)\) or \(P(X \geqslant 10)\) or \(P(X \gt 10)\) PI by 0.952 or 0.983 or 0.0172 or correct answer Allow figures to 2sf | M1 | 1.1a |
| Obtains correct probability ACF AWFW [0.0479, 0.048] | A1 | 1.1b |
| (2) |
Typical solution
(i)
0.00684
(ii)
\[P(X \leqslant 3) = 0.107\](iii)
\[P(X \geqslant 10) = 1 - P(X \leqslant 9)\]\[= 1 - 0.952\]\[= 0.048\]| Scheme | Marks | AO |
|---|---|---|
| Uses \(np(1 - p)\) | M1 | 1.1a |
| Forms a correct equation in \(p\) using 10 and 1.5 ACF | A1 | 3.1a |
| Obtains values for \(p\) ACF ISW Allow AWFW [0.34, 0.342] and [0.658, 0.66] for \(p\) or \(p = \dfrac{10 \pm \sqrt{10}}{20}\) | A1 | 1.1b |
| (3) | ||
| (10 marks) |