S2 June 2007 Q7

EdexcelOld spec12 marksBinomial Distribution Normal Distribution

(a)

(i) Write down two conditions for \(X \sim \mathrm{Bin}(n, p)\) to be approximated by a normal distribution \(Y \sim \mathrm{N}(\mu, \sigma^2)\). (2)

(ii) Write down the mean and variance of this normal approximation in terms of \(n\) and \(p\). (2)

A factory manufactures 2000 DVDs every day. It is known that 3% of DVDs are faulty.

(b) Using a normal approximation, estimate the probability that at least 40 faulty DVDs are produced in one day. (5)

The quality control system in the factory identifies and destroys every faulty DVD at the end of the manufacturing process. It costs £0.70 to manufacture a DVD and the factory sells non-faulty DVDs for £11.

Mark scheme (a)

Scheme	Marks
(i) If \(X \sim \mathrm{B}(n, p)\) and
\(n\) is large or \(n \gt 10\) or \(np \gt 5\) or \(nq \gt 5\)	B1
\(p\) is close to 0.5 or \(nq \gt 5\) and \(np \gt 5\)	B1
then \(X\) can be approximated by \(\mathrm{N}(np, np(1 - p))\)	(2)
(ii) mean \(= np\)	B1
variance \(= np(1 - p)\)	B1
	(2)

Notes

(ii) 2nd B1 must be in terms of \(p\)

Mark scheme (b)

Scheme	Marks
\(X \sim \mathrm{N}(60, 58.2)\) or \(X \sim \mathrm{N}(60, 7.63^2)\)	B1, B1
\(\mathrm{P}(X \geqslant 40) = \mathrm{P}(X \gt 39.5)\)	M1
\(= 1 - \mathrm{P}\left(z \lt \pm\left(\dfrac{39.5 - 60}{\sqrt{58.2}}\right)\right)\)	M1
\(= 1 - \mathrm{P}(z \lt -2.68715\ldots)\) \(= 0.9965\)	A1dep on both M
	(5)

Notes

B1, B1 60, 58.2

1st M1 using 39.5 or 40.5

2nd M1 standardising 39.5 or 40 or 40.5 and their \(\mu\) and \(\sigma\)

A1 allow answers in range 0.996 – 0.997

Scheme	Marks
\(\mathrm{E}(X) = 60\)	B1ft
Expected profit \(= (2000 - 60) \times 11 - 2000 \times 0.70\)	M1
= £19 940.	A1
	(3)
	(12 marks)