| Exam Board | Edexcel |
|---|---|
| Module | Paper 3 (Paper 3) |
| Year | 2022 |
| Session | June |
| Marks | 6 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Hypothesis test of binomial distributions |
| Type | State hypotheses only |
| Difficulty | Easy -1.2 Part (a) asks only to state hypotheses for a binomial test, which is pure recall requiring no calculation or problem-solving. Writing H₀: p = 0.1 and H₁: p ≠ 0.1 is a routine textbook exercise well below average A-level difficulty, though the full question involves more challenging critical region work. |
| Spec | 2.05b Hypothesis test for binomial proportion |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| \(H_0: p=0.1 \quad H_1: p\neq 0.1\) | B1 | For both hypotheses in terms of \(p\) or \(\pi\). Connected to \(H_0\) and \(H_1\) correctly. Condone 10% but not 10 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| Use of \(X\sim B(50,0.1)\) implied by sight of one of awrt 0.0052 or awrt 0.9755 or awrt 0.0245 | M1 | Using correct distribution to find the probability associated with one tail of the CR. If correct distribution is stated (may be seen in part(a)) allow for one tail of the correct CR or one of (awrt 0.025 or awrt 0.005 or awrt 0.975) seen connected to a correct probability statement |
| Critical regions \(X=0\) or \(X\geqslant 10\) | A1 | Lower CR \(X=0\ /\ X<1\ /\ X\leqslant 0/\) [condone eg \(P(X=0)\) labelled as CR]. Or Upper CR \(X\geqslant 10\) or \(X>9\) [condone \(P(X\geqslant 10)\) oe labelled as CR] |
| \(X=0\) and \(X\geqslant 10\) plus \(P(X=0)=\) awrt 0.0052 and \(P(X\geqslant 10)=\) awrt 0.0245 | A1 | Both CR's correct with relevant probabilities. Allow \(\cup\) for "and" and \(X>9\), \(X<1\), \(X\leqslant 0\). [do not allow \(P(X=0)\) or \(P(X\geqslant 10)\) oe]. Allow CR in different form eg \((9,\infty)\), \([10,\infty)\) |
| SC: Both CR correct with no probabilities and no distribution seen scores M0A1A0 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| 0.0297 | B1ft | awrt 0.0297 or 2.97% or ft for the sum of the probabilities in (b) for "their 2 critical regions" if seen. If none seen it must be awrt 0.0297. SC M0 in (b) for a one tail test: Allow B1ft for their one tail CR in (b) eg 0.0338 or 0.0245 or 0.0579 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| 15 is in the critical region therefore there is evidence to support the manager's belief | B1ft | A correct statement about 15 and "their CR" or sight \(P(X\geqslant 15)=0.0000738...\) and comparison with "their 0.0245" and a compatible correct statement in context. eg There is evidence that there has been a change in the proportion/probability arriving late. Condone increase rather than change. Do not allow contradicting statements. NB No CR given in (b) then B0 |
# Question 4:
## Part (a):
| Answer | Mark | Guidance |
|--------|------|----------|
| $H_0: p=0.1 \quad H_1: p\neq 0.1$ | B1 | For both hypotheses in terms of $p$ or $\pi$. Connected to $H_0$ and $H_1$ correctly. Condone 10% but not 10 |
## Part (b):
| Answer | Mark | Guidance |
|--------|------|----------|
| Use of $X\sim B(50,0.1)$ implied by sight of one of awrt 0.0052 or awrt 0.9755 or awrt 0.0245 | M1 | Using correct distribution to find the probability associated with one tail of the CR. If correct distribution is stated (may be seen in part(a)) allow for one tail of the correct CR **or** one of (awrt 0.025 or awrt 0.005 or awrt 0.975) seen connected to a correct probability statement |
| Critical regions $X=0$ or $X\geqslant 10$ | A1 | Lower CR $X=0\ /\ X<1\ /\ X\leqslant 0/$ [condone eg $P(X=0)$ labelled as CR]. Or Upper CR $X\geqslant 10$ or $X>9$ [condone $P(X\geqslant 10)$ oe labelled as CR] |
| $X=0$ and $X\geqslant 10$ plus $P(X=0)=$ awrt 0.0052 and $P(X\geqslant 10)=$ awrt 0.0245 | A1 | Both CR's correct with relevant probabilities. Allow $\cup$ for "and" and $X>9$, $X<1$, $X\leqslant 0$. [**do not allow** $P(X=0)$ or $P(X\geqslant 10)$ oe]. Allow CR in different form eg $(9,\infty)$, $[10,\infty)$ |
| SC: Both CR correct with no probabilities and no distribution seen scores M0A1A0 | | |
## Part (c):
| Answer | Mark | Guidance |
|--------|------|----------|
| 0.0297 | B1ft | awrt 0.0297 or 2.97% or ft for the sum of the probabilities in (b) for "their 2 critical regions" if seen. If none seen it must be awrt 0.0297. SC M0 in (b) for a one tail test: Allow B1ft for their one tail CR in (b) eg 0.0338 or 0.0245 or 0.0579 |
## Part (d):
| Answer | Mark | Guidance |
|--------|------|----------|
| 15 is in the critical region therefore there is evidence to support the **manager**'s belief | B1ft | A correct statement about 15 and "their CR" or sight $P(X\geqslant 15)=0.0000738...$ and comparison with "their 0.0245" **and** a compatible correct statement in context. eg There is evidence that there has been a change in the **proportion/probability** arriving **late**. Condone increase rather than change. Do not allow contradicting statements. NB No CR given in (b) then B0 |
---\begin{enumerate}
\item A dentist knows from past records that $10 \%$ of customers arrive late for their appointment.
\end{enumerate}
A new manager believes that there has been a change in the proportion of customers who arrive late for their appointment.
A random sample of 50 of the dentist's customers is taken.\\
(a) Write down
\begin{itemize}
\item a null hypothesis corresponding to no change in the proportion of customers who arrive late
\item an alternative hypothesis corresponding to the manager's belief\\
(b) Using a $5 \%$ level of significance, find the critical region for a two-tailed test of the null hypothesis in (a) You should state the probability of rejection in each tail, which should be less than 0.025\\
(c) Find the actual level of significance of the test based on your critical region from part (b)
\end{itemize}
The manager observes that 15 of the 50 customers arrived late for their appointment.\\
(d) With reference to part (b), comment on the manager's belief.
\hfill \mbox{\textit{Edexcel Paper 3 2022 Q4 [6]}}