S2 January 2007 Q4

EdexcelOld spec12 marksPoisson Distribution

(a) State the condition under which the normal distribution may be used as an approximation to the Poisson distribution. (1)

(b) Explain why a continuity correction must be incorporated when using the normal distribution as an approximation to the Poisson distribution. (1)

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.

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. (6)

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. (2)

Scheme	Marks
\(\lambda \gt 10\) or large	B1
	(1)

Scheme	Marks
The Poisson is discrete and the normal is continuous.	B1
	(1)

Mark scheme (c)

Scheme	Marks
Let \(Y\) represent the number of yachts hired in winter \(\mathrm{P}(Y \lt 3) = \mathrm{P}(Y \leqslant 2)\)	M1
\(= 0.1247\)	A1
	(2)

M1 \(\mathrm{P}(Y \leqslant 2)\) & Po(5)

A1 awrt 0.125

Mark scheme (d)

Scheme	Marks
Let \(X\) represent the number of yachts hired in summer \(X \sim \mathrm{Po}(25)\). \(\mathrm{N}(25, 25)\)	B1
\(\mathrm{P}(X \gt 30) \approx \mathrm{P}\left(Z \gt \dfrac{30.5 - 25}{5}\right)\)	M1;M1
\(\approx \mathrm{P}(Z \gt 1.1)\)	A1
\(\approx 1 - 0.8643\)	M1
\(\approx 0.1357\)	A1
	(6)

B1 all correct, can be implied by standardisation below

M1;M1 \(\pm\) standardise with 25 & 5; \(\pm 0.5\) c.c.

1st A1 1.1

3rd M1 ‘one minus’

2nd A1 awrt 0.136