Edexcel S2 2023 October — Question 5 16 marks

Exam BoardEdexcel
ModuleS2 (Statistics 2)
Year2023
SessionOctober
Marks16
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Mark schemeDownload PDF ↗
TopicHypothesis test of a Poisson distribution
TypeOne-tailed test (increase or decrease)
DifficultyStandard +0.3 This is a standard S2 hypothesis testing question covering routine Poisson distribution calculations and tests. Parts (a)-(b) are straightforward recall and calculation, part (c) is a textbook one-tailed test, and part (d) adds a normal approximation which is a standard S2 technique. All steps follow predictable patterns with no novel problem-solving required, making it slightly easier than average.
Spec2.04d Normal approximation to binomial2.05b Hypothesis test for binomial proportion2.05c Significance levels: one-tail and two-tail5.02i Poisson distribution: random events model5.02j Poisson formula: P(X=x) = e^(-lambda)*lambda^x/x!5.02k Calculate Poisson probabilities5.02l Poisson conditions: for modelling
  1. A supermarket receives complaints at a mean rate of 6 per week.
    1. State one assumption necessary, in order for a Poisson distribution to be used to model the number of complaints received by the supermarket.
    2. Find the probability that, in a given week, there are
      1. fewer than 3 complaints received by the supermarket,
      2. at least 6 complaints received by the supermarket.
    In a randomly selected week, the supermarket received 12 complaints.
  2. Test, at the \(5 \%\) level of significance, whether or not there is evidence that the mean number of complaints is greater than 6 per week.
    State your hypotheses clearly. Following changes made by the supermarket, it received 26 complaints over a 6-week period.
  3. Use a suitable approximation to test whether or not there is evidence that, following the changes, the mean number of complaints received is less than 6 per week. You should state your hypotheses clearly and use a 5\% significance level.
Question 5:
Part (a)
AnswerMarks Guidance
AnswerMark Guidance
Complaints received are independent or occurring at a constant rate or singlyB1 Must be in context: need 'complaints' and then independent/random or constant rate or singly
Part (b)(i)
AnswerMarks Guidance
AnswerMark Guidance
\([P(X<3 \mid X\sim\text{Po}(6))=]\ 0.0620\)B1 awrt 0.062
Part (b)(ii)
AnswerMarks Guidance
AnswerMark Guidance
\([P(X\geq 6)=]\ 1-P(X\leq 5)\) or \(1-0.4457 = 0.5543\)M1A1 M1 for writing or using \(1-P(X\leq 5)\). A1 awrt 0.554
Part (c)
AnswerMarks Guidance
AnswerMark Guidance
\(H_0: \lambda=6\), \(H_1: \lambda>6\)B1 Both hypotheses correct. Must be attached to \(H_0\) and \(H_1\) in terms of \(\lambda\) or \(\mu\)
\(P(X\geq 12)=1-P(X\leq 11)=[1-0.9799]\) or \(P(X\geq 11)=1-P(X\leq 10)=[1-0.9574]\)M1 For writing or using \(1-P(X\leq 11)\) or \(1-P(X\leq 10)\)
\(= 0.0201\) or \(\text{CR}\geq 11\)A1 For 0.0201 or \(\text{CR}\geq 11\)
Reject \(H_0\) / In the CR / SignificantM1 A correct statement — no context needed but do not allow contradicting non-contextual comments
There is sufficient evidence to suggest that the mean number of complaints received is greater than 6 per weekA1ft Correct conclusion in context with highlighted words
Part (d)
AnswerMarks Guidance
AnswerMark Guidance
\(H_0: \lambda=6\), \(H_1: \lambda<6\)B1 Both hypotheses correct. Allow use of 36 rather than 6
6 week period is \(\text{Po}(36)\Rightarrow N(36,36)\)B1 For writing or using \(N(36,36)\)
\(P(Y\leq 26)\approx P\!\left(Z<\frac{26.5-36}{6}\right)\) or \(\frac{x+0.5-36}{\sqrt{36}}<-1.6449\)M1 M1 M1 for standardising using 25.5/26/26.5 with correct mean and sd; M1 for correct continuity correction e.g. 26.5 or \(x+0.5\)
\([P(Z<-1.583...)]=0.0571\) or \(x<25.63...\) awrt 0.057 / awrt 25.6A1 awrt 0.057 (NB Poisson gives 0.0512685… scores M0M0A0) or \(\text{CR}<\) awrt 25.6
Do not reject \(H_0\) / Not in the CR / Not significantM1 A correct statement — no context needed
There is insufficient evidence to suggest that the mean number of complaints received after the changes is less than 6 per weekA1ft Correct conclusion in context with highlighted words. Allow "stayed the same/not changed" oe 
# Question 5:

## Part (a)
| Answer | Mark | Guidance |
|--------|------|----------|
| Complaints received are independent or occurring at a constant rate or singly | B1 | Must be in context: need 'complaints' and then independent/random or constant rate or singly |

## Part (b)(i)
| Answer | Mark | Guidance |
|--------|------|----------|
| $[P(X<3 \mid X\sim\text{Po}(6))=]\ 0.0620$ | B1 | awrt 0.062 |

## Part (b)(ii)
| Answer | Mark | Guidance |
|--------|------|----------|
| $[P(X\geq 6)=]\ 1-P(X\leq 5)$ or $1-0.4457 = 0.5543$ | M1A1 | M1 for writing or using $1-P(X\leq 5)$. A1 awrt 0.554 |

## Part (c)
| Answer | Mark | Guidance |
|--------|------|----------|
| $H_0: \lambda=6$, $H_1: \lambda>6$ | B1 | Both hypotheses correct. Must be attached to $H_0$ and $H_1$ in terms of $\lambda$ or $\mu$ |
| $P(X\geq 12)=1-P(X\leq 11)=[1-0.9799]$ or $P(X\geq 11)=1-P(X\leq 10)=[1-0.9574]$ | M1 | For writing or using $1-P(X\leq 11)$ or $1-P(X\leq 10)$ |
| $= 0.0201$ or $\text{CR}\geq 11$ | A1 | For 0.0201 or $\text{CR}\geq 11$ |
| Reject $H_0$ / In the CR / Significant | M1 | A correct statement — no context needed but do not allow contradicting non-contextual comments |
| There is sufficient evidence to suggest that the mean **number** of **complaints** received is **greater** than 6 per week | A1ft | Correct conclusion in context with highlighted words |

## Part (d)
| Answer | Mark | Guidance |
|--------|------|----------|
| $H_0: \lambda=6$, $H_1: \lambda<6$ | B1 | Both hypotheses correct. Allow use of 36 rather than 6 |
| 6 week period is $\text{Po}(36)\Rightarrow N(36,36)$ | B1 | For writing or using $N(36,36)$ |
| $P(Y\leq 26)\approx P\!\left(Z<\frac{26.5-36}{6}\right)$ or $\frac{x+0.5-36}{\sqrt{36}}<-1.6449$ | M1 M1 | M1 for standardising using 25.5/26/26.5 with correct mean and sd; M1 for correct continuity correction e.g. 26.5 or $x+0.5$ |
| $[P(Z<-1.583...)]=0.0571$ or $x<25.63...$ awrt 0.057 / awrt 25.6 | A1 | awrt 0.057 (NB Poisson gives 0.0512685… scores M0M0A0) or $\text{CR}<$ awrt 25.6 |
| Do not reject $H_0$ / Not in the CR / Not significant | M1 | A correct statement — no context needed |
| There is insufficient evidence to suggest that the mean **number** of **complaints** received after the changes is **less** than 6 per week | A1ft | Correct conclusion in context with highlighted words. Allow "stayed the same/not changed" oe |

---
\begin{enumerate}
  \item A supermarket receives complaints at a mean rate of 6 per week.\\
(a) State one assumption necessary, in order for a Poisson distribution to be used to model the number of complaints received by the supermarket.\\
(b) Find the probability that, in a given week, there are\\
(i) fewer than 3 complaints received by the supermarket,\\
(ii) at least 6 complaints received by the supermarket.
\end{enumerate}

In a randomly selected week, the supermarket received 12 complaints.\\
(c) Test, at the $5 \%$ level of significance, whether or not there is evidence that the mean number of complaints is greater than 6 per week.\\
State your hypotheses clearly.

Following changes made by the supermarket, it received 26 complaints over a 6-week period.\\
(d) Use a suitable approximation to test whether or not there is evidence that, following the changes, the mean number of complaints received is less than 6 per week. You should state your hypotheses clearly and use a 5\% significance level.

\hfill \mbox{\textit{Edexcel S2 2023 Q5 [16]}}
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