| Exam Board | Edexcel |
|---|---|
| Module | AS Paper 2 (AS Paper 2) |
| Year | 2023 |
| Session | June |
| Marks | 7 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Hypothesis test of binomial distributions |
| Type | Two-tailed test critical region |
| Difficulty | Standard +0.3 This is a standard two-tailed binomial hypothesis test with straightforward application of critical region methodology. While it requires understanding of hypothesis testing concepts and binomial probability calculations, it follows a routine template with no novel problem-solving required. The multi-part structure and need to find critical regions using tables makes it slightly above average difficulty for AS-level, but remains a textbook-style question. |
| Spec | 2.05a Hypothesis testing language: null, alternative, p-value, significance2.05b Hypothesis test for binomial proportion2.05c Significance levels: one-tail and two-tail |
| Answer | Marks | Guidance |
|---|---|---|
| \([H_1:]\, p \neq 0.25\) | B1 | Correct alternative hypothesis; may be stated in terms of \(p\) or \(\pi\); ignore null hypothesis if stated |
| Answer | Marks | Guidance |
|---|---|---|
| \(X \sim B(50,\ 0.25)\) | B1 | Setting up Binomial model with \(n=50\) and \(p=0.25\) (allow if seen previously); may be implied by M mark |
| Answer | Marks | Guidance |
|---|---|---|
| or \(X \leq 6\) or \(X \geq 19\) | M1 | Use of \(B(50, 0.25)\) to find a tail probability or CR tail. May be implied by relevant probability e.g. \(P(X \leq 7)=0.0453\), \(P(X \geq 19)=0.986\), \(P(X \geq 20)=0.0139\) |
| \([P(X \leq 6) =]\,\) awrt \(0.0194\) and \([P(X \geq 19) =]\,\) awrt \(0.0287\) | A1 | Both correct probabilities seen (condone awrt 0.0193 and awrt 0.0288) |
| CR: \(X \leq 6\) or \(X \geq 19\) | A1 | Correct CR; e.g. \(X < 7\), \(X > 18\); condone \(X \leq 6\) and \(X \geq 19\) |
| Answer | Marks | Guidance |
|---|---|---|
| \([0.0194 + 0.0287 =]\,\) awrt \(0.048\) | B1ft | awrt 0.048 or ft their two-tailed CR from \(B(50,p)\) to 2sf accuracy; each tail probability must be \(< 0.05\) |
| Answer | Marks | Guidance |
|---|---|---|
| (Do not reject \(H_0\),) there is insufficient evidence to suggest that the proportion of those with the allergy differs from 25%/Rylan's belief not supported | B1 | Correct inference in context. Allow 'proportion'/'probability'/'percent(age)/%' but not 'number'. 'Rylan's hypothesis is not supported' is B1, but 'Rylan's hypothesis test is not supported' is B0. |
## Question 4:
**Part (a):**
$[H_1:]\, p \neq 0.25$ | B1 | Correct alternative hypothesis; may be stated in terms of $p$ or $\pi$; ignore null hypothesis if stated
**Part (b):**
$X \sim B(50,\ 0.25)$ | B1 | Setting up Binomial model with $n=50$ and $p=0.25$ (allow if seen previously); may be implied by M mark
$[P(X \leq 6) =]\, 0.0194$ or $[P(X \geq 18) =]\, 0.9713$ or $[P(X \geq 19) =]\, 0.0287$
or $X \leq 6$ or $X \geq 19$ | M1 | Use of $B(50, 0.25)$ to find a tail probability or CR tail. May be implied by relevant probability e.g. $P(X \leq 7)=0.0453$, $P(X \geq 19)=0.986$, $P(X \geq 20)=0.0139$
$[P(X \leq 6) =]\,$ awrt $0.0194$ and $[P(X \geq 19) =]\,$ awrt $0.0287$ | A1 | Both correct probabilities seen (condone awrt 0.0193 and awrt 0.0288)
CR: $X \leq 6$ or $X \geq 19$ | A1 | Correct CR; e.g. $X < 7$, $X > 18$; condone $X \leq 6$ and $X \geq 19$
**Part (c):**
$[0.0194 + 0.0287 =]\,$ awrt $0.048$ | B1ft | awrt 0.048 or ft their two-tailed CR from $B(50,p)$ to 2sf accuracy; each tail probability must be $< 0.05$
**Part (d):**
(Do not reject $H_0$,) there is insufficient evidence to suggest that the **proportion** of those with the **allergy** differs from 25%/**Rylan's belief** not supported | B1 | Correct inference in context. Allow 'proportion'/'probability'/'percent(age)/%' but not 'number'. 'Rylan's hypothesis is not supported' is B1, but 'Rylan's hypothesis test is not supported' is B0.
---\begin{enumerate}
\item Past information shows that $25 \%$ of adults in a large population have a particular allergy.
\end{enumerate}
Rylan believes that the proportion that has the allergy differs from 25\%\\
He takes a random sample of 50 adults from the population.\\
Rylan carries out a test of the null hypothesis $\mathrm { H } _ { 0 } : p = 0.25$ using a $5 \%$ level of significance.\\
(a) Write down the alternative hypothesis for Rylan's test.\\
(b) Find the critical region for this test.
You should state the probability associated with each tail, which should be as close to $2.5 \%$ as possible.\\
(c) State the actual probability of incorrectly rejecting $\mathrm { H } _ { 0 }$ for this test.
Rylan finds that 10 of the adults in his sample have the allergy.\\
(d) State the conclusion of Rylan's hypothesis test.
\hfill \mbox{\textit{Edexcel AS Paper 2 2023 Q4 [7]}}