S4 June 2005 Q5

EdexcelOld spec13 marksIncludes hypothesis testingQuality of Tests

5. Define

(a) a Type I error, (1)

(b) the size of a test. (1)

Jane claims that she can read Alan’s mind. To test this claim Alan randomly chooses a card with one of 4 symbols on it. He then concentrates on the symbol. Jane then attempts to read Alan’s mind by stating what symbol she thinks is on the card. The experiment is carried out 8 times and the number of times \(X\) that Jane is correct is recorded.

The probability of Jane stating the correct symbol is denoted by \(p\).

To test the hypothesis \(\mathrm{H}_0 : p = 0.25\) against \(\mathrm{H}_1 : p \gt 0.25\), a critical region of \(X \gt 6\) is used.

(d) Show that the power function of this test is \(8p^7 - 7p^8\). (3)

Given that \(p = 0.3\), calculate

(e) the power of this test, (1)

(f) the probability of a Type II error. (2)

(g) Suggest two ways in which you might reduce the probability of a Type II error. (2)

Scheme	Marks
A Type I error occurs when \(\mathrm{H}_0\) is rejected when in fact it is true.	B1
	(1)

Scheme	Marks
The size of a test is the probability of a Type I error	B1
	(1)

Scheme	Marks
\(X \sim \mathrm{B}(8, 0.25)\) Can be implied	B1
Size \(= \mathrm{P}(X \gt 6) = 1 - \mathrm{P}(X \leqslant 6 \mid n = 8,\ p = 0.25)\)	M1
\(= 1 - 0.9996\) \(= 0.0004\)	A1
	(3)

Mark scheme (d)

Scheme	Marks
Power \(= \mathrm{P}(X \gt 6 \mid p = p,\ n = 8)\) \(= \mathrm{P}(X = 7) + \mathrm{P}(X = 8)\)	M1
\(= \dfrac{8!}{7!\,1!}p^7(1 - p) + p^8\)	A1
\(= 8p^7 - 8p^8 + p^8\) \(= 8p^7 - 7p^8\) *	A1
	(3)

Scheme	Marks
Power when \(p = 0.3 = 8 \times 0.3^7 - 7 \times 0.3^8\) \(= 0.00129\) awrt 0.0013	B1
	(1)

Scheme	Marks
\(\mathrm{P}(\text{Type II error}) = 1 - \text{Power}(0.3)\)	M1
\(= 0.99870\ldots\)	A1
	(2)