Question 3 - A-Level Maths

Edexcel S2 2014 June — Question 3 13 marks

Exam Board	Edexcel
Module	S2 (Statistics 2)
Year	2014
Session	June
Marks	13
Paper	Download PDF ↗
Mark scheme	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 Poisson hypothesis testing question with routine procedures: stating model assumptions, finding critical regions from tables, and conducting tests. Part (b) requires table lookup for two-tailed critical values, and part (e) is a one-tailed test with parameter scaling (15 minutes → mean of 6). All steps follow textbook methods with no novel problem-solving required, making it slightly easier than average.
Spec	5.02i Poisson distribution: random events model 5.02j Poisson formula: P(X=x) = e^(-lambda)*lambda^x/x!5.02k Calculate Poisson probabilities 5.05c Hypothesis test: normal distribution for population mean

A company claims that it receives emails at a mean rate of 2 every 5 minutes.
1. Give two reasons why a Poisson distribution could be a suitable model for the number of emails received.
2. Using a \(5 \%\) level of significance, find the critical region for a two-tailed test of the hypothesis that the mean number of emails received in a 10 minute period is 4 . The probability of rejection in each tail should be as close as possible to 0.025
3. Find the actual level of significance of this test.
To test this claim, the number of emails received in a random 10 minute period was recorded. During this period 8 emails were received.
Comment on the company's claim in the light of this value. Justify your answer. During a randomly selected 15 minutes of play in the Wimbledon Men's Tennis Tournament final, 2 emails were received by the company.
Test, at the \(10 \%\) level of significance, whether or not the mean rate of emails received by the company during the Wimbledon Men's Tennis Tournament final is lower than the mean rate received at other times. State your hypotheses clearly.

Show mark scheme Show mark scheme source

Question 3:

Part (a)

Answer	Marks	Guidance
Answer/Working	Marks	Guidance
Any two of: Emails are independent/occur at random; Emails occur singly; Emails occur at a constant rate	B1B1d	B1: any correct statement with context of emails. B1d: dependent on previous B1, any correct statement, need not have context. SC: 2 correct statements without context B1 B0

Part (b)

Answer	Marks	Guidance
Answer/Working	Marks	Guidance
\(X \sim Po(4)\)
\(P(X=0) = 0.0183\)	B1	\(X=0\) or \(X \leq 0\). Allow any letter
\(P(X \geq 9) = 0.0214\)	B1	\(X \geq 9\) or \(X > 8\). Allow any letter. SC: if correct CRs written as probability statements award B1 B0
CR: \(X=0\); \(X \geq 9\)

Part (c)

Answer	Marks	Guidance
Answer/Working	Marks	Guidance
\(0.0183 + 0.0214 = 0.0397\) or \(3.97\%\)	M1A1	M1: adding their probabilities of their critical regions if sum gives probability less than 1, or award if correct answer given. A1: awrt 0.0397

Part (d)

Answer	Marks	Guidance
Answer/Working	Marks	Guidance
8 is not in the critical region or \(P(X \geq 8) = 0.0511\)	M1	Correct reason ft their CR. Do not allow non-contextual contradictions
Therefore there is evidence that the company's claim is true	A1ft	Correct conclusion for their CR. Allow conclusion in context of emails received at a rate of 2 every 5 mins

Part (e)

Answer	Marks	Guidance
Answer/Working	Marks	Guidance
\(H_0: \lambda = 6\) (or \(\lambda=2\)) \(\quad H_1: \lambda < 6\) (or \(\lambda=2\))	B1	Both hypotheses correct; must have \(\lambda\) or \(\mu\) and either 2 or 6
\(Po(6)\)	M1	Using \(Po(6)\); may be implied by correct answer
\(P(X \leq 2) = 0.0620\) \(\quad\) CR: \(X \leq 2\)	A1	0.062 or \(X \leq 2\)
\(0.0620 < 0.10\); Reject \(H_0\) or Significant	M1dep	Dependent on previous method being awarded. Do not allow conflicting non-contextual statements. Follow through their hypotheses
There is evidence at the 10% level of significance that the mean rate/number/amount of emails received is lower/has decreased/is less. Or: fewer emails are received	A1cso

# Question 3:

## Part (a)

| Answer/Working | Marks | Guidance |
|---|---|---|
| Any two of: Emails are independent/occur at random; Emails occur singly; Emails occur at a constant rate | B1B1d | B1: any correct statement with context of emails. B1d: dependent on previous B1, any correct statement, need not have context. SC: 2 correct statements without context B1 B0 |

## Part (b)

| Answer/Working | Marks | Guidance |
|---|---|---|
| $X \sim Po(4)$ | | |
| $P(X=0) = 0.0183$ | B1 | $X=0$ or $X \leq 0$. Allow any letter |
| $P(X \geq 9) = 0.0214$ | B1 | $X \geq 9$ or $X > 8$. Allow any letter. SC: if correct CRs written as probability statements award B1 B0 |
| CR: $X=0$; $X \geq 9$ | | |

## Part (c)

| Answer/Working | Marks | Guidance |
|---|---|---|
| $0.0183 + 0.0214 = 0.0397$ or $3.97\%$ | M1A1 | M1: adding their probabilities of their critical regions if sum gives probability less than 1, or award if correct answer given. A1: awrt 0.0397 |

## Part (d)

| Answer/Working | Marks | Guidance |
|---|---|---|
| 8 is not in the critical region **or** $P(X \geq 8) = 0.0511$ | M1 | Correct reason ft their CR. Do not allow non-contextual contradictions |
| Therefore there is evidence that the company's **claim** is true | A1ft | Correct conclusion for their CR. Allow conclusion in context of **emails** received at a rate of 2 every 5 mins |

## Part (e)

| Answer/Working | Marks | Guidance |
|---|---|---|
| $H_0: \lambda = 6$ (or $\lambda=2$) $\quad H_1: \lambda < 6$ (or $\lambda=2$) | B1 | Both hypotheses correct; must have $\lambda$ or $\mu$ and either 2 or 6 |
| $Po(6)$ | M1 | Using $Po(6)$; may be implied by correct answer |
| $P(X \leq 2) = 0.0620$ $\quad$ CR: $X \leq 2$ | A1 | 0.062 or $X \leq 2$ |
| $0.0620 < 0.10$; Reject $H_0$ or Significant | M1dep | Dependent on previous method being awarded. Do not allow conflicting non-contextual statements. Follow through their hypotheses |
| There is evidence at the 10% level of significance that the mean **rate/number/amount** of **emails** received **is lower/has decreased/is less**. Or: **fewer emails** are received | A1cso | |

---

Show LaTeX source

\begin{enumerate}
  \item A company claims that it receives emails at a mean rate of 2 every 5 minutes.\\
(a) Give two reasons why a Poisson distribution could be a suitable model for the number of emails received.\\
(b) Using a $5 \%$ level of significance, find the critical region for a two-tailed test of the hypothesis that the mean number of emails received in a 10 minute period is 4 . The probability of rejection in each tail should be as close as possible to 0.025\\
(c) Find the actual level of significance of this test.
\end{enumerate}

To test this claim, the number of emails received in a random 10 minute period was recorded.

During this period 8 emails were received.\\
(d) Comment on the company's claim in the light of this value. Justify your answer.

During a randomly selected 15 minutes of play in the Wimbledon Men's Tennis Tournament final, 2 emails were received by the company.\\
(e) Test, at the $10 \%$ level of significance, whether or not the mean rate of emails received by the company during the Wimbledon Men's Tennis Tournament final is lower than the mean rate received at other times. State your hypotheses clearly.\\

\hfill \mbox{\textit{Edexcel S2 2014 Q3 [13]}}

This paper (6 questions)

View full paper

Q1 8 Q2 14 Q3 13 Q4 14 Q5 11 Q6 15