S2 June 2012 Q3

EdexcelOld spec9 marksIncludes hypothesis testingPoisson Distribution

(a) Write down two conditions needed to approximate the binomial distribution by the Poisson distribution. (2)

A machine which manufactures bolts is known to produce 3% defective bolts. The machine breaks down and a new machine is installed. A random sample of 200 bolts is taken from those produced by the new machine and 12 bolts were defective.

(b) Using a suitable approximation, test at the 5% level of significance whether or not the proportion of defective bolts is higher with the new machine than with the old machine. State your hypotheses clearly. (7)

Scheme	Marks
\(n\) – large/high/big/ \(n \gt 50\)	B1
\(p\) – small/close to 0 / \(p \lt 0.2\)	B1
	(2)

Mark scheme (b)

Scheme	Marks
\(\mathrm{H}_0 : p = 0.03 \qquad \mathrm{H}_1 : p \gt 0.03\)	B1,B1
Po(6)	B1
\(\mathrm{P}(X \geqslant 12) = 1 - \mathrm{P}(X \leqslant 11)\) or \(\mathrm{P}(X \leqslant 10) = 0.9574\)	M1
\(= 1 - 0.9799\) \(\mathrm{P}(X \geqslant 11) = 0.0426\)
\(= 0.0201\) CR \(X \geqslant 11\)	A1
\((0.0201 \lt 0.05)\) Reject \(\mathrm{H}_0\) or Significant or 12 lies in the Critical region.	M1 dep.
There is evidence that the proportion of defective bolts has increased.	A1 ft
	(7)
	(9 marks)

Notes

1st B1 for \(\mathrm{H}_0 : p = 0.03\)

2nd B1 for \(\mathrm{H}_1 : p \gt 0.03\)

SC If both hypotheses are correct but a different letter to \(p\) is used they get B1 B0

Also allow B1 B0 for \(\mathrm{H}_0 : \lambda = 6\) and \(\mathrm{H}_1 : \lambda \gt 6\)

B1 writing or using Po(6)

One tail

1st M1 for writing or using \(1 - \mathrm{P}(X \leqslant 11)\) or giving \(\mathrm{P}(X \leqslant 10) = 0.9574\) or giving \(\mathrm{P}(X \geqslant 11) = 0.0426\). May be implied by correct CR or probability = 0.0201

1st A1 for 0.0201 or CR \(X \geqslant 11\)/ \(X \gt 10\). NB \(\mathrm{P}(X \leqslant 11) = 0.9799\) on its own scores M1A1

2nd M1 dependent on the 1st M1 being awarded. For a correct statement based on the table below. Do not allow non-contextual conflicting statements eg “significant” and “accept \(\mathrm{H}_0\)”. Ignore comparisons.

2nd A1 ft for a correct contextualised statement. NB A correct contextual statement on its own scores M1A1.

	\(0.05 \lt p \lt 0.95\)	\(p \lt 0.05\) or \(p \gt 0.95\)
2nd M1	not significant/ accept \(\mathrm{H}_0\)/ Not in CR	significant/ reject \(\mathrm{H}_0\)/ In CR
2nd A1	The proportion/number/amount/percentage oe of defective bolts has not increased/is not higher/oe	The proportion/number/amount/percentage oe of defective bolts has increased/is higher/oe

Two tail

1st M1 for writing or using \(1 - \mathrm{P}(X \leqslant 11)\) or giving \(\mathrm{P}(X \geqslant 12) = 0.0201\) or giving \(\mathrm{P}(X \leqslant 11) = 0.9799\). May be implied by correct CR or probability = 0.0201

1st A1 for 0.0201 or CR \(X \geqslant 12\)/ \(X \gt 11\). NB \(\mathrm{P}(X \leqslant 11) = 0.9799\) on its own scores M1A1

2nd A1 ft for a correct contextualised statement. NB A correct contextual statement on its own scores M1A1.

	\(0.025 \lt p \lt 0.975\)	\(p \lt 0.025\) or \(p \gt 0.975\)
2nd M1	not significant/ accept \(\mathrm{H}_0\)/ Not in CR	significant/ reject \(\mathrm{H}_0\)/ In CR
2nd A1	The proportion/number/amount/percentage oe of defective bolts has not increased/is not higher/oe	The proportion/number/amount/percentage oe of defective bolts has increased/is higher/oe

Use of N(6,5.82) May get B1 B1 B0 M1 (must use 11.5)A0 M1dep A1 ft