S2 January 2008 Q6

EdexcelOld spec12 marksBinomial Distribution Normal Distribution

6. The probability that a sunflower plant grows over 1.5 metres high is 0.25. A random sample of 40 sunflower plants is taken and each sunflower plant is measured and its height recorded.

(a) Find the probability that the number of sunflower plants over 1.5 m high is between 8 and 13 (inclusive) using

(i) a Poisson approximation,

(ii) a Normal approximation. (10)

(b) Write down which of the approximations used in part (a) is the most accurate estimate of the probability. You must give a reason for your answer. (2)

Mark scheme (a)

Scheme	Marks
(i) Let \(X\) represent the number of sunflower plants more than 1.5m high \(X \sim \mathrm{Po}(10)\) \(\mu = 10\)	B1
\(\mathrm{P}(8 \leqslant X \leqslant 13) = \mathrm{P}(X \leqslant 13) - \mathrm{P}(X \leqslant 7)\) \(= 0.8645 - 0.2202\)	M1
\(= 0.6443\) awrt 0.644	A1
(ii) \(X \sim \mathrm{N}(10, 7.5)\)	B1
\(\mathrm{P}(7.5 \leqslant X \leqslant 13.5) = \mathrm{P}\left(\dfrac{7.5 - 10}{\sqrt{7.5}} \leqslant X \leqslant \dfrac{13.5 - 10}{\sqrt{7.5}}\right)\)	M1 M1 A1 A1
\(= \mathrm{P}(-0.913 \leqslant X \leqslant 1.278)\) \(= 0.8997 - (1 - 0.8186)\)	M1
\(= 0.7183\) awrt 0.718 or 0.719	A1
	(10)

Notes

(i) B1 mean = 10 May be implied in (i) or (ii)

M1 Attempting to find \(\mathrm{P}(X \leqslant 13) - \mathrm{P}(X \leqslant 7)\)

A1 awrt 0.644

(ii) B1 \(\sigma^2 = 7.5\) May be implied by being correct in standardised formula

M1 using 7.5 or 8.5 or 12.5 or 13.5.

M1 standardising using 7.5 or 8 or 8.5 or 12.5 or 13 or 13.5 and their mean and standard deviation.

A1 award for either \(\dfrac{7.5 - 10}{\sqrt{7.5}}\) or awrt \(-0.91\)

A1 award for either \(\dfrac{13.5 - 10}{\sqrt{7.5}}\) or awrt 1.28

M1 Finding the correct area. Following on from their 7.5 and 13.5. Need to do a Prob >0.5 – prob <0.5 or prob <0.5 + prob< 0.5

A1 awrt 0.718 or 0.719 only. Dependent on them getting all three method marks.

No working but correct answer will gain all the marks

Mark scheme (b)

Scheme	Marks
Normal approx /not Poisson	B1
since (\(n\) is large) and \(p\) close to half. or (\(np = 10\) \(npq = 7.5\)) mean \(\ne\) variance or \(np\) (= 10) and \(nq\) (= 30) both >5. or exact binomial = 0.7148	B1dep
	(2)
	(12 marks)

Notes

first B1 normal

second B1
\(p\) close to half,
or mean \(\ne\) variance
or \(np\) and \(nq\) both > 5. They may use a number bigger than 5
or they may work out the exact value 0.7148 using the binomial distribution.

Do not allow \(np \gt 5\) and \(npq \gt 5\)