S4 June 2013 (R) Q7

EdexcelOld spec9 marksIncludes hypothesis testingQuality of Tests

7. A machine produces bricks. The lengths, \(x\) mm, of the bricks are distributed \(\mathrm{N}(\mu, 2^2)\).
At the start of each week a random sample of \(n\) bricks is taken to check the machine is working correctly.
A test is then carried out at the 1% level of significance with

\[\mathrm{H}_0 : \mu = 202 \quad \text{and} \quad \mathrm{H}_1 : \mu \lt 202\]

(a) Find, in terms of \(n\), the critical region of the test. (3)

The probability of a type II error, when \(\mu = 200\), is less than 0.05

(b) Find the minimum value of \(n\). (6)

Mark scheme (a)

Scheme	Marks
\(\dfrac{CV - 202}{2/\sqrt{n}} = -2.3263\)	M1 B1
CR \(\leqslant 202 - \dfrac{4.6526}{\sqrt{n}}\) or \(202 - 2.3263\sqrt{\dfrac{4}{n}}\)	A1
	(3)

Notes

Note only lose one B1 for not reading from points table. This should be deducted the first time it is done

1st M1 use correct formula equal a \(z\) value

A1 allow use of <

Mark scheme (b)

Scheme	Marks
\(\dfrac{CV - 200}{2/\sqrt{n}} = 1.6449\) or \(\dfrac{2 - \dfrac{4.6526}{\sqrt{n}}}{\dfrac{2}{\sqrt{n}}} \gt 1.6449\)	M1 B1
\(\text{CV} = 200 + \dfrac{3.2898}{\sqrt{n}}\)
Solving simultaneously
\(2 = \dfrac{7.9424}{\sqrt{n}}\) or \(\sqrt{n} - \dfrac{4.6526}{2} \gt 1.6449\)	M1
\(\sqrt{n} = 3.9712\)	A1
\(n = 15.77\)	A1
\(n = 16\)	A1
	(6)
	(9 marks)

Notes

1st M1 use correct formula equal a \(z\) value

B1 – if B mark lost in part (a) allow 1.64 or 1.65

1st A1 awrt 3.97 may be implied by an answer of 15.77 or an answer of 16 and using 1.6449

2nd A1 awrt 15.8 may be implied by an answer of 16