S4 June 2009 Q3

EdexcelOld spec12 marksIncludes hypothesis testingQuality of Tests

3. Define, in terms of \(\mathrm{H}_0\) and/or \(\mathrm{H}_1\),

(a) the size of a hypothesis test, (1)

(b) the power of a hypothesis test. (1)

The probability of getting a head when a coin is tossed is denoted by \(p\).

This coin is tossed 12 times in order to test the hypotheses \(\mathrm{H}_0 : p = 0.5\) against \(\mathrm{H}_1 : p \neq 0.5\), using a 5% level of significance.

(d) Given that \(p = 0.4\)

(i) find the probability of a type II error when using this test,

(ii) find the power of this test. (4)

(e) Suggest two ways in which the power of the test can be increased. (2)

Scheme	Marks
Size is the probability of \(\mathrm{H}_0\) being rejected when it is in fact true. or \(\mathrm{P}(\text{reject }\mathrm{H}_0 / \mathrm{H}_0\text{ is true})\) oe	B1
	(1)

Mark scheme (b)

Scheme	Marks
The power of the test is the probability of rejecting \(\mathrm{H}_0\) when \(\mathrm{H}_1\) is true. or \(\mathrm{P}(\text{rejecting }\mathrm{H}_0/\mathrm{H}_1\text{ is true})\) / \(\mathrm{P}(\text{rejecting }\mathrm{H}_0/\mathrm{H}_0\text{ is false})\) oe	B1
	(1)

Mark scheme (c)

Scheme	Marks
\(X \sim \mathrm{B}(12, 0.5)\)	B1
\(\mathrm{P}(X \leqslant 2) = 0.0193\)	M1
\(\mathrm{P}(X \geqslant 10) = 0.0193\)
\(\therefore\) critical region is \(\{X \leqslant 2 \cup X \geqslant 10\}\)	A1A1
	(4)

Mark scheme (d)

Scheme	Marks
(i) \(\mathrm{P}(\text{Type II error}) = \mathrm{P}(3 \leqslant X \leqslant 9 \mid p = 0.4)\)	M1
\(= \mathrm{P}(X \leqslant 9) - \mathrm{P}(X \leqslant 2)\) \(= 0.9972 - 0.0834\)	M1dep
\(= 0.9138\)	A1
(ii) Power \(= 1 - 0.9138\) \(= 0.0862\)	B1 ft
	(4)

Notes

(d) (i) first M1 for either correct area or follow through from their critical region
2nd M1 dependent on them having the first M1. for finding their area correctly
A1 cao

(ii) B1 follow through from their (i)

Scheme	Marks
Increase the sample size	B1
Increase the significance level/larger critical region	B1
	(2)
	(12 marks)