| Exam Board | Edexcel |
|---|---|
| Module | AS Paper 2 (AS Paper 2) |
| Year | 2021 |
| Session | November |
| Marks | 10 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Hypothesis test of binomial distributions |
| Type | Multiple binomial probability calculations |
| Difficulty | Standard +0.3 This is a straightforward AS-level binomial hypothesis test question with standard parts: stating the distribution, calculating probabilities using given parameters, and conducting a two-tailed test at 5% significance. All steps are routine applications of textbook methods with no novel problem-solving required, making it slightly easier than average. |
| Spec | 2.04b Binomial distribution: as model B(n,p)2.04c Calculate binomial probabilities2.05b Hypothesis test for binomial proportion2.05c Significance levels: one-tail and two-tail |
| Answer | Marks | Guidance |
|---|---|---|
| \([R = \text{no. of red beads in Aliya's bracelet}]\ R \sim B(18, 0.14)\) | B1 | Accept in words e.g. binomial with \(n = 18\) and \(p = 0.14\) |
| Answer | Marks | Guidance |
|---|---|---|
| \(P(R = 1) = 0.19403\ldots\) awrt 0.194 | B1 | For awrt 0.194 |
| Answer | Marks | Guidance |
|---|---|---|
| \(P(R \ldots 4) = 1 - P(R \leq 3) = 1 - [0.76184\ldots]\) | M1 | For interpreting "at least 4" Need \(1 - P(R \leq 3)\) and \(1 - p\ [0 < p < 1]\ P(R=3) = 0.233\).. OK |
| \(= 0.2381588\ldots\) awrt 0.238 | A1 | For awrt 0.238 |
| Answer | Marks | Guidance |
|---|---|---|
| Requires \(p = 0.14\) to be constant so need a large number of beads in the sack to ensure that removing 18 beads does not appreciably affect this probability, then it could be suitable. | B1 | For mention of large number of beads and need for \(p = 0.14\) to be constant for it to be suitable. Do NOT accept e.g. "events are independent" |
| Answer | Marks | Guidance |
|---|---|---|
| \(H_0: p = 0.14 \quad H_1: p \neq 0.14\) | B1 | For both hypotheses correct with use of \(p\) or \(\pi\) |
| \([X = \text{number of red beads in the sample}]\ X \sim B(75, 0.14)\) | M1 | For selecting a suitable model: sight or correct use of \(B(75, 0.14)\). May be implied by sight of 0.015 or better or \([P(X > 4)] = 0.9849\ldots\) i.e. 0.985 or better |
| \(P(X \leq 4) = 0.01506\ldots\) or if \(B(75, 0.14)\) seen awrt 0.02 | A1 | 1st A1 for use of correct model awrt 0.015 (accept awrt 0.02 following a correct expression). Allow 1st A1 for awrt 0.985 only if correct comparison with 0.975 is seen. Sight of \(B(75, 0.14)\) and \(P(X \leq 4) =\) awrt 0.02 scores M1A1 |
| \(\{0.02 < 0.025\) so significant or reject \(H_0\}\) There is evidence that the proportion of red beads has changed | A1 | 2nd A1 (dep on M1A1) for a correct conclusion in context mentioning "proportion", "red" and "changed". If there is a statement about \(H_0\) or significance it must be compatible. |
| Answer | Marks | Guidance |
|---|---|---|
| \(p\text{-value is } 2\times 0.01506\ldots = 0.030123\ldots =\) awrt 0.03 | B1ft | For awrt 0.03. Allow ft of their probability in (d) provided at least 3sf used. NB an answer of 0.02 in (d) leading to 0.04 in (e) is B0. Use of CR will give significance level of \(0.01506\ldots + 0.01406\ldots = 0.029\ldots\) score B1 no ft |
# Question 4:
## Part (a)
$[R = \text{no. of red beads in Aliya's bracelet}]\ R \sim B(18, 0.14)$ | B1 | Accept in words e.g. binomial with $n = 18$ and $p = 0.14$
## Part (b)(i)
$P(R = 1) = 0.19403\ldots$ awrt **0.194** | B1 | For awrt 0.194
## Part (b)(ii)
$P(R \ldots 4) = 1 - P(R \leq 3) = 1 - [0.76184\ldots]$ | M1 | For interpreting "at least 4" Need $1 - P(R \leq 3)$ and $1 - p\ [0 < p < 1]\ P(R=3) = 0.233$.. OK
$= 0.2381588\ldots$ awrt **0.238** | A1 | For awrt 0.238
## Part (c)
Requires $p = 0.14$ to be constant so need a large number of beads in the sack to ensure that removing 18 beads does not appreciably affect this probability, then it could be suitable. | B1 | For mention of large number of beads and need for $p = 0.14$ to be constant for it to be suitable. Do NOT accept e.g. "events are independent"
## Part (d)
$H_0: p = 0.14 \quad H_1: p \neq 0.14$ | B1 | For both hypotheses correct with use of $p$ or $\pi$
$[X = \text{number of red beads in the sample}]\ X \sim B(75, 0.14)$ | M1 | For selecting a suitable model: sight or correct use of $B(75, 0.14)$. May be implied by sight of 0.015 or better or $[P(X > 4)] = 0.9849\ldots$ i.e. 0.985 or better
$P(X \leq 4) = 0.01506\ldots$ or if $B(75, 0.14)$ seen awrt 0.02 | A1 | 1st A1 for use of correct model awrt 0.015 (accept awrt 0.02 following a correct expression). Allow 1st A1 for awrt 0.985 only if correct comparison with 0.975 is seen. Sight of $B(75, 0.14)$ and $P(X \leq 4) =$ awrt 0.02 scores M1A1
$\{0.02 < 0.025$ so significant or reject $H_0\}$ There is evidence that the proportion of red beads has changed | A1 | 2nd A1 (dep on M1A1) for a correct conclusion in context mentioning "proportion", "red" and "changed". If there is a statement about $H_0$ or significance it must be compatible.
## Part (e)
$p\text{-value is } 2\times 0.01506\ldots = 0.030123\ldots =$ awrt 0.03 | B1ft | For awrt 0.03. Allow ft of their probability in (d) provided at least 3sf used. NB an answer of 0.02 in (d) leading to 0.04 in (e) is B0. Use of CR will give significance level of $0.01506\ldots + 0.01406\ldots = 0.029\ldots$ score B1 **no ft**
---\begin{enumerate}
\item A nursery has a sack containing a large number of coloured beads of which $14 \%$ are coloured red.
\end{enumerate}
Aliya takes a random sample of 18 beads from the sack to make a bracelet.\\
(a) State a suitable binomial distribution to model the number of red beads in Aliya's bracelet.\\
(b) Use this binomial distribution to find the probability that\\
(i) Aliya has just 1 red bead in her bracelet,\\
(ii) there are at least 4 red beads in Aliya's bracelet.\\
(c) Comment on the suitability of a binomial distribution to model this situation.
After several children have used beads from the sack, the nursery teacher decides to test whether or not the proportion of red beads in the sack has changed. She takes a random sample of 75 beads and finds 4 red beads.\\
(d) Stating your hypotheses clearly, use a 5\% significance level to carry out a suitable test for the teacher.\\
(e) Find the $p$-value in this case.
\hfill \mbox{\textit{Edexcel AS Paper 2 2021 Q4 [10]}}