| Exam Board | Edexcel |
|---|---|
| Module | Paper 3 (Paper 3) |
| Year | 2023 |
| Session | June |
| Marks | 9 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Hypothesis test of binomial distributions |
| Type | State test assumptions or distributions |
| Difficulty | Moderate -0.8 This is a straightforward hypothesis testing question with standard binomial distribution calculations. Part (a) requires stating a basic condition (independence), parts (b) and (c) are routine binomial probability calculations, and part (d) is a standard one-tailed hypothesis test with clearly stated parameters. All steps follow textbook procedures with no novel insight required, making it easier than average. |
| Spec | 2.04b Binomial distribution: as model B(n,p)2.04c Calculate binomial probabilities5.05b Unbiased estimates: of population mean and variance |
| Answer | Marks | Guidance |
|---|---|---|
| Comment about 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 (1) | May use idea of independent events: suitable reason in context covering random packing or packets filled independently. Must see key words. B0 for idea of only 2 cases or idea of a fixed number of trials. |
| Answer | Marks | Guidance |
|---|---|---|
| \([P(T=6) =]\ 0.17273\ldots\) awrt \(\mathbf{0.173}\) | B1 | For awrt 0.173. |
| Answer | Marks | Guidance |
|---|---|---|
| \([P(T < 3) = P(T,\ 2) =]\ 0.061587\ldots\) awrt \(\mathbf{0.0616}\) | B1 (2) | For awrt 0.0616. |
| Answer | Marks | Guidance |
|---|---|---|
| \(K \sim B(5,\ \text{"0.0616"})\) | M1 | For sight of \(B(5, \text{"0.0616"})\) or \({}^5C_2(\text{"0.0616"})^2(1-\text{"0.0616"})^3\) ft their answer to (b)(ii). |
| \(P(K=2) = 0.031344\ldots\) in range \([\mathbf{0.0313}\text{--}\mathbf{0.0314}]\) | A1 (2) | For answer in range [0.0313 to 0.0314]. Use of 0.0616 gives 0.031356, ans only 2/2. |
| Answer | Marks | Guidance |
|---|---|---|
| \(H_0: p = \tfrac{1}{7} \quad H_1: p < \tfrac{1}{7}\) | B1 | For both hypotheses correct in terms of \(p\) or \(\pi\). |
| \([X = \text{no. of packets containing a prize}]\ X \sim B\!\left(110,\ \tfrac{1}{7}\right)\) | M1 | For selecting an appropriate model, may be implied by 1st A1 or \(P(X=9) = 0.0199(2\ldots)\). Also allow Normal: sight of \(N\!\left(\frac{110}{7}, \frac{660}{49}\right)\) or awrt 13.5, or probability 0.045(20..) or 0.033(66..). |
| \([P(X \leqslant 9)] = 0.038292\ldots\) | A1 | 1st A1 for 0.038 or better, or allow 0.04 with sight of \(P(X,\ 9)\). ALT Critical Region: Allow CR of \(X \leqslant 9\) (or \(X < 10\)) provided supporting probability seen. |
| [Significant result or reject \(H_0\)] e.g. there is evidence to support Kamil's claim | A1 (4) | 2nd A1 (dep on 1st A1 and indep of hyp's) for suitable conclusion in context that suggests support for Kamil's claim or states evidence that proportion/probability/chance of packets containing a prize is less than \(\frac{1}{7}\). Do not award 2nd A1 for contradictory statements. |
# Question 2:
## Part (a)
Comment about **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 **(1)** | May use idea of independent events: suitable reason in context covering random packing or packets filled independently. Must see key words. B0 for idea of only 2 cases or idea of a fixed number of trials.
## Part (b)(i)
$[P(T=6) =]\ 0.17273\ldots$ awrt $\mathbf{0.173}$ | B1 | For awrt 0.173.
## Part (b)(ii)
$[P(T < 3) = P(T,\ 2) =]\ 0.061587\ldots$ awrt $\mathbf{0.0616}$ | B1 **(2)** | For awrt 0.0616.
## Part (c)
$[K = \text{no. of boxes with fewer than 3 packets containing a prize}]$
$K \sim B(5,\ \text{"0.0616"})$ | M1 | For sight of $B(5, \text{"0.0616"})$ or ${}^5C_2(\text{"0.0616"})^2(1-\text{"0.0616"})^3$ ft their answer to (b)(ii).
$P(K=2) = 0.031344\ldots$ in range $[\mathbf{0.0313}\text{--}\mathbf{0.0314}]$ | A1 **(2)** | For answer in range [0.0313 to 0.0314]. Use of 0.0616 gives 0.031356, ans only 2/2.
## Part (d)
$H_0: p = \tfrac{1}{7} \quad H_1: p < \tfrac{1}{7}$ | B1 | For both hypotheses correct in terms of $p$ or $\pi$.
$[X = \text{no. of packets containing a prize}]\ X \sim B\!\left(110,\ \tfrac{1}{7}\right)$ | M1 | For selecting an appropriate model, may be implied by 1st A1 or $P(X=9) = 0.0199(2\ldots)$. Also allow Normal: sight of $N\!\left(\frac{110}{7}, \frac{660}{49}\right)$ or awrt 13.5, or probability 0.045(20..) or 0.033(66..).
$[P(X \leqslant 9)] = 0.038292\ldots$ | A1 | 1st A1 for 0.038 or better, or allow 0.04 with sight of $P(X,\ 9)$. **ALT Critical Region:** Allow CR of $X \leqslant 9$ (or $X < 10$) provided supporting probability seen.
[Significant result or reject $H_0$] e.g. there is evidence to support Kamil's claim | A1 **(4)** | 2nd A1 (dep on 1st A1 and indep of hyp's) for suitable conclusion in context that suggests support for Kamil's claim **or** states evidence that proportion/probability/chance of packets containing a prize is less than $\frac{1}{7}$. Do not award 2nd A1 for contradictory statements.
---\begin{enumerate}
\item A machine fills packets with sweets and $\frac { 1 } { 7 }$ of the packets also contain a prize.
\end{enumerate}
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) State a condition needed for $T$ to be modelled by $\mathrm { B } \left( 40 , \frac { 1 } { 7 } \right)$
A box is selected at random.\\
(b) Using $T \sim \mathrm {~B} \left( 40 , \frac { 1 } { 7 } \right)$ find\\
(i) the probability that the box has exactly 6 packets containing a prize,\\
(ii) the probability that the box has fewer than 3 packets containing a prize.
Kamil's sweet shop buys 5 boxes of these sweets.\\
(c) Find the probability that exactly 2 of these 5 boxes have fewer than 3 packets containing a prize.
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.\\
(d) Use a suitable test to assess Kamil's claim.
You should
\begin{itemize}
\item state your hypotheses clearly
\item use a $5 \%$ level of significance
\end{itemize}
\hfill \mbox{\textit{Edexcel Paper 3 2023 Q2 [9]}}