| Exam Board | Edexcel |
|---|---|
| Module | S2 (Statistics 2) |
| Year | 2024 |
| Session | October |
| Paper | Download PDF ↗ |
| Topic | Hypothesis test of a Poisson distribution |
| Type | Two-tailed test setup or execution |
| Difficulty | Standard +0.3 This is a standard S2 hypothesis testing question following a routine template: identify distribution, state assumptions, set up hypotheses, find critical region for two-tailed test, calculate actual significance level, and interpret result. All steps are textbook procedures with no novel problem-solving required, though it does require careful handling of discrete distribution and two-tailed critical regions, making it slightly above average difficulty. |
| Spec | 2.05a Hypothesis testing language: null, alternative, p-value, significance5.02i Poisson distribution: random events model |
| Answer | Marks | Guidance |
|---|---|---|
| (a) | Po(7) | B1 |
| (b) | Customers enter the shop occur singly/randomly/independently/constant (average) rate | B1, B1 |
| (c) | \(H_0: \lambda = '7'\) \(H_1: \lambda \neq '7'\) | B1ft |
| (d) | \(P(X_r, \mathbf{D}) = \text{awrt } 0.0073\) and \(P(X_r, 2) = \text{awrt } 0.0296\) | M1 |
| \(P(X...\mathbf{13}) = \text{awrt } 0.0270\) and \(P(X...14) = \text{awrt } 0.0128\) | M1 | |
| \(X_r, \mathbf{D} \cup X ...14\) | A1 | (3) |
| (e) | \(0.0073 + 0.0128 = 0.0201\) So 2.01% | M1 |
| A1ft | (2) | |
| (f) | 12 is not in the critical region So, there is insufficient evidence that rate of customers entering the shop has changed | M1 |
| A1 | (2) | |
| Total | 11 |
| (a) | Po(7) | B1 | (1) |
|---|---|---|---|
| (b) | Customers enter the shop occur singly/randomly/independently/**constant** (average) rate | B1, B1 | (2) |
| (c) | $H_0: \lambda = '7'$ $H_1: \lambda \neq '7'$ | B1ft | (1) |
| (d) | $P(X_r, \mathbf{D}) = \text{awrt } 0.0073$ and $P(X_r, 2) = \text{awrt } 0.0296$ | M1 | |
| | $P(X...\mathbf{13}) = \text{awrt } 0.0270$ and $P(X...14) = \text{awrt } 0.0128$ | M1 | |
| | $X_r, \mathbf{D} \cup X ...14$ | A1 | (3) |
| (e) | $0.0073 + 0.0128 = 0.0201$ So 2.01% | M1 | |
| | | A1ft | (2) |
| (f) | 12 is not in the critical region So, there is insufficient evidence that **rate of customers entering the shop has changed** | M1 | |
| | | A1 | (2) |
| | **Total** | **11** | |
**Notes for Question 3:**
- **(a) B1:** Correct distribution fully specified. Po(Poisson) and $\lambda = 7$
- **(b) B1, B1:** For two of the given assumptions (must have context of customers/people)
- Context only needs to be stated once. (B1B0 for one assumption in context or for two assumptions with no context)
- **(c) B1ft:** Both hypotheses correct. Must be attached to $H_0$ and $H_1$ in terms of $\lambda$ or $\mu$; Ft their 7 from part (a) in the hypotheses
- **(d) M1:** Use of Po(7) to find the lower critical value. May be implied by either awrt 0.0073 or awrt 0.0296 seen (must be seen in part (d))
- Also implied by $X = 1$ or $X_r, \mathbf{D}$
- Use of Po(7) to find the upper critical value.
- May be implied by awrt 0.0270 or awrt 0.0128 or awrt 0.973 or awrt 0.987 seen (must be seen in part (d))
- Also implied by $X = 14$ or $X ...14$
- **(d) M1:** $X_r, \mathbf{D} \cup X ...14$ correct CR scores 3 out of 3 but $14_r, X_r, \mathbf{D}$ is M1M1A0
- Allow equivalent forms e.g. $X < 2, X > 13$
- Must be a CR and not a probability statement
- $P(X_r, \mathbf{D}), P(X...14)$ scores M1M1A0
- **(e) M1:** Adding the two probabilities (each must be less than 0.05) for their critical region
- **(e) A1ft:** awrt 0.0201 or awrt 2.01% ft the sum of their two selected probability tails
- For a correct comparison of 12 with their CR (or their implied CR if one is not explicitly stated),
- 12 is not in the CR condone $12 < '14'$
- Finding $P(X = 12)$ is M0
- Finding $P(X ...12)$ on its own is M0, they must state 12 is not in the CR
- **(f) A1:** Correct conclusion in context. Must be a rate, e.g. number in/per 10-minute period (not number on its own).
- No hypotheses in part (c) then A0
- Do not allow comments about the manager's claim on its own, e.g. The manager's claim is not supported.
- This is not a ft mark.
---\begin{enumerate}
\item During Monday afternoons, customers are known to enter a certain shop at a mean rate of 7 customers every 10 minutes.\\
(a) Suggest a suitable distribution to model the number of customers that enter this shop in a 10-minute interval on Monday afternoons.\\
(b) State two assumptions necessary for this distribution to be a suitable model of this situation.
\end{enumerate}
A new shop manager wants to find out if the rate of customers has changed since they took over.\\
(c) Write down suitable null and alternative hypotheses that the shop manager should use.
The shop manager decides to monitor the number of customers entering the shop in a random 10-minute interval next Monday afternoon.\\
(d) Using a $3 \%$ level of significance, find the critical region to test whether the rate of customers has changed.\\
(e) Find the actual significance level of this test based on your critical region from part (d)
During the random 10-minute interval that Monday afternoon, 12 customers entered the shop.\\
(f) Comment on this finding, using the critical region in part (d)
\hfill \mbox{\textit{Edexcel S2 2024 Q3}}