June 2023 Paper 3 Q2
2. A machine fills packets with sweets and \(\frac{1}{7}\) of the packets also contain a prize.
The packets of sweets are placed in boxes before being delivered to shops.
There are 40 packets of sweets in each box.
The random variable \(T\) represents the number of packets of sweets that contain a prize in each box.
A box is selected at random.
Kamil’s sweet shop buys 5 boxes of these sweets.
Kamil claims that the proportion of packets containing a prize is less than \(\frac{1}{7}\)
A random sample of 110 packets is taken and 9 packets contain a prize.
You should
- state your hypotheses clearly
- use a 5% level of significance
| Scheme | Marks | AO |
|---|---|---|
| Comment in context about either independence or random packing e.g. “prizes must be placed in packets at random/independently of each other” or about constant probability e.g. “the probability of a packet containing a prize is constant/ the same/fixed” | B1 | 3.5b |
| (1) |
Notes
B1: May use idea of independent events: a suitable reason, in context, covering idea of random packing or packets filled independently.
Should mention key words/ideas of: prizes in packets or packets in boxes
May use idea of constant probability. Must see key words underlined in scheme.
Idea of probability with “independence” or “not affected by other packets” is B0
B0 for: Idea of only 2 cases. E.g. Packet contains a prize or not
or Idea of a fixed number of trials. E.g. Need a fixed number of packets in each box
| Scheme | Marks | AO |
|---|---|---|
| (i) \([\mathrm{P}(T = 6) =]\) 0.17273… awrt 0.173 | B1 | 1.1b |
| (ii) \([\mathrm{P}(T \lt 3) = \mathrm{P}(T \leqslant 2) =]\) 0.061587… awrt 0.0616 | B1 | 1.1b |
| (2) |
Notes
(b)(i) B1: for awrt 0.173
(ii) B1: for awrt 0.0616
| Scheme | Marks | AO |
|---|---|---|
| [\(K\) = no. of boxes with fewer than 3 packets containing a prize] \(K \sim \mathrm{B}(5,\ \text{``}0.0616\text{''})\) | M1 | 1.1b |
| \(\mathrm{P}(K = 2) = 0.031344\ldots\) in the range [0.0313~0.0314] | A1 | 1.1b |
| (2) |
Notes
M1: for sight of \(\mathrm{B}(5,\ \text{``}0.0616\text{''})\) or \({}^{5}C_{2}(\text{``}0.0616\text{''})^2(1 - \text{``}0.0616\text{''})^3\) ft their answer to (b)(ii).
A1: for an answer in the range [0.0313 to 0.0314] Use of 0.0616 gives 0.031356..ans only 2/2
| Scheme | Marks | AO |
|---|---|---|
| \(\mathrm{H}_0 : p = \tfrac{1}{7} \qquad \mathrm{H}_1 : p \lt \tfrac{1}{7}\) | B1 | 2.5 |
| [\(X\) = no of packets containing a prize] \(X \sim \mathrm{B}\left(110,\ \tfrac{1}{7}\right)\) | M1 | 3.3 |
| \([\mathrm{P}(X \leqslant 9)] = 0.038292\ldots\) | A1 | 3.4 |
| [Significant result or reject \(\mathrm{H}_0\)] E.g. there is evidence to support Kamil’s claim | A1 | 2.2b |
| (4) | ||
| (9 marks) |
Notes
B1: for both hypotheses correct in terms of \(p\) or \(\pi\)
M1: for selecting an appropriate model, may be implied by 1st A1 or \(\mathrm{P}(X = 9) = 0.0199(2\ldots)\)
1st A1: for 0.038 or better or allow 0.04 with sight of \(\mathrm{P}(X \leqslant 9)\)
ALT Critical Region. Allow CR of \(X \leqslant 9\) (or \(X \lt 10\)) provided a supporting probability is seen
e.g. A1 for correct CR plus \(\mathrm{P}(X \leqslant 10) = 0.0718\ldots\) (accept 2sf or 1sf if prob statement seen)
2nd A1: (dep on 1st A1 but indep of hyp’s) for a suitable conclusion in context that suggests support for (Kamil’s) claim or states that there is evidence that proportion /probability/chance of packets containing a prize is less than \(\tfrac{1}{7}\)
Do not award 2nd A1 for contradictory statements e.g. “not significant” so “supports claim”
Normal: Sight of \(\mathrm{N}\left(\dfrac{110}{7},\ \dfrac{660}{49} \text{ or awrt } 13.5\right)\) or probability of 0.045(20..) or 0.033(66..) scores M1