S3 June 2006 Q8

EdexcelOld spec13 marksIncludes hypothesis testingChi-Squared Tests

8. Five coins were tossed 100 times and the number of heads recorded. The results are shown in the table below.

Number of heads	0	1	2	3	4	5
Frequency	6	18	29	34	10	3

(a) Suggest a suitable distribution to model the number of heads when five unbiased coins are tossed. (2)

(b) Test, at the 10% level of significance, whether or not the five coins are unbiased. State your hypotheses clearly. (11)

Scheme	Marks
\(\mathrm{B}(5, 0.5)\)	M1 A1
	(2)

Mark scheme (b)

Scheme

Marks

\(\mathrm{H}_0 : \mathrm{B}(5, 0.5)\) is a suitable model (good fit)
\(\mathrm{H}_1 : \mathrm{B}(5, 0.5)\) is not a suitable model (not a good fit)
ft for \(\hat{p} = 0.466\).

B1ft

No. of heads	0	1	2	3	4	5
Expected	3.125	15.625	31.25	31.25	15.625	3.125
Actual	6	18	29	34	10	3

\(100\,\mathrm{P}(X = x)\) for Bin, 1 correct = A1, All correct = A1, 3sf or better

M1 A1 A1

	\(O\)	\(E\)	\(\dfrac{(O - E)^2}{E}\)
0 or 1	24	18.75	1.47
2	29	31.25	0.162
3	34	31.25	0.242
4 or 5	13	18.75	\(1.76\dot{3}\)

grouped \(O\) and \(E\). All correct 2sf or better.

M1 A1

\(\displaystyle\sum \frac{(O - E)^2}{E} = 3.637\dot{3}\) \(\sum\) required, awrt 3.64

M1 A1

\(\nu = 4 - 1 = 3\), \(\chi^2_3(0.10) = 6.251\)

B1 B1ft

Insufficient evidence to reject \(\mathrm{H}_0\)
\(\mathrm{B}(5, 0.5)\) is a suitable model.
No evidence that coins are biased

A1ft

(11)

(13 marks)

Notes

Ungrouped gives awrt 5.44, \(\nu = 5\), \(\chi^2_5 = 9.236\)