S4 June 2016 Q4

EdexcelOld spec9 marksIncludes hypothesis testingQuality of Tests

4. A manufacturer produces boxes of screws containing short screws and long screws. The manufacturer claims that the probability, \(p\), of a randomly selected screw being long, is 0.5

A shopkeeper does not believe the manufacturer’s claim. He designs two tests, \(A\) and \(B\), to test the hypotheses \(\mathrm{H}_0 : p = 0.5\) and \(\mathrm{H}_1 : p \lt 0.5\)

In test \(A\), a random sample of 10 screws is taken from a box of screws and \(\mathrm{H}_0\) is rejected if there are fewer than 3 long screws.

In test \(B\), a random sample of 5 screws is taken from a box of screws and \(\mathrm{H}_0\) is rejected if there are no long screws, otherwise a second random sample of 5 screws is taken from a box of screws. If there are no long screws in this second sample \(\mathrm{H}_0\) is rejected, otherwise it is accepted.

(a) Find the size of test \(A\). (1)

(b) Find the size of test \(B\). (3)

Some values, to 2 decimal places, of the power function for test \(A\) and the power function for test \(B\) are given in the table below.

\(p\)	0.1	0.2	0.3	0.4
Power test \(A\)	0.93	\(r\)	0.38	0.17
Power test \(B\)	0.83	0.55	0.31	0.15

(d) Find the value of \(r\). (1)

The shopkeeper believes that the value of \(p\) is less than 0.4

(e) Suggest which of the tests the shopkeeper should use. Give a reason for your answer. (2)

Scheme	Marks
Size of test \(A = \mathrm{P}(Y \leqslant 2)\) \(= 0.0547\)	B1
	(1)

Mark scheme (b)

Scheme	Marks
Size of test \(B = \mathrm{P}(\text{Rejecting } \mathrm{H}_0 \mid p = 0.5)\) \(= \mathrm{P}(X = 0) + (1 - \mathrm{P}(X = 0)) \times \mathrm{P}(X = 0)\)	M1
\(= 0.5^5 + (1 - 0.5^5)(0.5^5)\)	A1
\(= 0.03125 + (0.96875)(0.03125)\) \(= 0.0615/0.0614\)	A1
	(3)

Notes

M1 for a correct expression/selection of probabilities

A1 for a correct expression in terms of probabilities. Allow 0.0312 + (0.9688)(0.0312)

Mark scheme (c)

Scheme	Marks
Power function of test \(B\) = P(0 long screws in first 5) + P(0 long screws in second 5 \| > 0 long screws in first 5) \(= \mathrm{P}(X = 0 \mid p) + [1 - \mathrm{P}(X = 0 \mid p)]\,\mathrm{P}(X = 0 \mid p)\)	M1
\(= (1 - p)^5 + [1 - (1 - p)^5](1 - p)^5\) \(= 2(1 - p)^5 - (1 - p)^{10}\)	A1
	(2)

Notes

M1 for a correct expression
A1 for a correct expression in terms of \(p\)

Scheme	Marks
Test \(A\) as it is more powerful for values of \(p \lt 0.4\)	M1 A1
	(2)
	(9 marks)