Edexcel S2 2007 January — Question 4 12 marks

Exam BoardEdexcel
ModuleS2 (Statistics 2)
Year2007
SessionJanuary
Marks12
PaperDownload PDF ↗
Mark schemeDownload PDF ↗
TopicApproximating the Poisson to the Normal distribution
TypeExplain continuity correction necessity
DifficultyModerate -0.8 This is a straightforward S2 question testing standard bookwork (parts a,b), routine Poisson calculation (part c), and standard normal approximation with continuity correction (parts d,e). All parts follow textbook procedures with no problem-solving insight required, making it easier than average A-level maths questions.
Spec2.04d Normal approximation to binomial2.04e Normal distribution: as model N(mu, sigma^2)2.04f Find normal probabilities: Z transformation
4. (a) State the condition under which the normal distribution may be used as an approximation to the Poisson distribution.
(b) Explain why a continuity correction must be incorporated when using the normal distribution as an approximation to the Poisson distribution. A company has yachts that can only be hired for a week at a time. All hiring starts on a Saturday.
During the winter the mean number of yachts hired per week is 5 .
(c) Calculate the probability that fewer than 3 yachts are hired on a particular Saturday in winter. During the summer the mean number of yachts hired per week increases to 25 . The company has only 30 yachts for hire.
(d) Using a suitable approximation find the probability that the demand for yachts cannot be met on a particular Saturday in the summer. In the summer there are 16 Saturdays on which a yacht can be hired.
(e) Estimate the number of Saturdays in the summer that the company will not be able to meet the demand for yachts.
Part (a)
AnswerMarks Guidance
\(\lambda > 10\) or large\(\mu\) ok B1 (1 mark)
Part (b)
AnswerMarks Guidance
The Poisson is discrete and the normal is continuous. B1 (1 mark)
Part (c)
AnswerMarks Guidance
Let \(Y\) represent the number of yachts hired in winter
\(P(Y < 3) = P(Y \leq 2)\)\(P(Y \leq 2)\) & Po(5) M1
\(= 0.1247\)awrt 0.125 A1 (2 marks)
Part (d)
AnswerMarks Guidance
Let \(X\) represent the number of yachts hired in summer \(X \sim \text{Po}(25)\).
\(N(25, 25)\) all correct, can be implied by standardisation belowB1
\(P(X > 30) = P\left(Z > \frac{30.5 - 25}{5}\right)\)\(\pm\) standardise with 25 & 5; \(\pm 0.5\) c.c. M1;M1
\(= P(Z > 1.1)\)1.1 A1
\(\approx 1 - 0.8643\)'one minus' M1
\(\approx 0.1357\)awrt 0.136 A1 (6 marks)
Part (e)
AnswerMarks Guidance
no. of weeks \(= 0.1357 \times 16\)ANS (d)\(\times 16\) M1
\(= 2.17\) or 2 or 3ans>16 M0A0 A1† (2 marks) 
**Part (a)**
$\lambda > 10$ or large | $\mu$ ok | B1 (1 mark) |

**Part (b)**
The Poisson is discrete and the normal is continuous. | | B1 (1 mark) |

**Part (c)**
Let $Y$ represent the number of yachts hired in winter | |
$P(Y < 3) = P(Y \leq 2)$ | $P(Y \leq 2)$ & Po(5) | M1 |
$= 0.1247$ | awrt 0.125 | A1 (2 marks) |

**Part (d)**
Let $X$ represent the number of yachts hired in summer $X \sim \text{Po}(25)$. | |
$N(25, 25)$ all correct, can be implied by standardisation below | B1 |
$P(X > 30) = P\left(Z > \frac{30.5 - 25}{5}\right)$ | $\pm$ standardise with 25 & 5; $\pm 0.5$ c.c. | M1;M1 |
$= P(Z > 1.1)$ | 1.1 | A1 |
$\approx 1 - 0.8643$ | 'one minus' | M1 |
$\approx 0.1357$ | awrt 0.136 | A1 (6 marks) |

**Part (e)**
no. of weeks $= 0.1357 \times 16$ | ANS (d)$\times 16$ | M1 |
$= 2.17$ or 2 or 3 | ans>16 M0A0 | A1† (2 marks) |

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4. (a) State the condition under which the normal distribution may be used as an approximation to the Poisson distribution.\\
(b) Explain why a continuity correction must be incorporated when using the normal distribution as an approximation to the Poisson distribution.

A company has yachts that can only be hired for a week at a time. All hiring starts on a Saturday.\\
During the winter the mean number of yachts hired per week is 5 .\\
(c) Calculate the probability that fewer than 3 yachts are hired on a particular Saturday in winter.

During the summer the mean number of yachts hired per week increases to 25 . The company has only 30 yachts for hire.\\
(d) Using a suitable approximation find the probability that the demand for yachts cannot be met on a particular Saturday in the summer.

In the summer there are 16 Saturdays on which a yacht can be hired.\\
(e) Estimate the number of Saturdays in the summer that the company will not be able to meet the demand for yachts.

\hfill \mbox{\textit{Edexcel S2 2007 Q4 [12]}}
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