S3 June 2009 Q5

EdexcelOld spec12 marksIncludes hypothesis testingChi-Squared Tests Poisson Distribution

5. The number of goals scored by a football team is recorded for 100 games. The results are summarised in Table 1 below.

Number of goals	Frequency
0	40
1	33
2	14
3	8
4	5

Table 1

(a) Calculate the mean number of goals scored per game. (2)

The manager claimed that the number of goals scored per match follows a Poisson distribution. He used the answer in part (a) to calculate the expected frequencies given in Table 2.

Number of goals	Expected Frequency
0	34.994
1	\(r\)
2	\(s\)
3	6.752
\(\geqslant 4\)	2.221

Table 2

(b) Find the value of \(r\) and the value of \(s\) giving your answers to 3 decimal places. (3)

Mark scheme (a)

Scheme	Marks
\(\lambda = \dfrac{0 \times 40 + 1 \times 33 + 2 \times 14 + 3 \times 8 + 4 \times 5}{100} = 1.05\)	M1 A1
	(2)

Notes

M1 for an attempt to find the mean- at least 2 terms on numerator seen
Correct answer only will score both marks

Mark scheme (b)

Scheme	Marks
Using Expected frequency \(= 100 \times \mathrm{P}(X = x) = 100 \times \dfrac{\mathrm{e}^{-1.05}1.05^x}{x!}\) gives	M1
\(r = 36.743\) awrt 36.743 or 36.744	A1
\(s = 19.290\) 19.29 or awrt 19.290	A1
	(3)

Notes

M1 for use of correct formula (ft their mean). 1^st A1 for \(r\), 2^nd A1 for \(s\) (19.29 OK)

Mark scheme (c)

Scheme

Marks

\(\mathrm{H}_0\) : Poisson distribution is a suitable model
\(\mathrm{H}_1\) : Poisson distribution is not a suitable model

Number of goals	Frequency	Expected frequency
0	40	34.994
1	33	36.743
2	14	19.290
3	8	6.752	8.972443
\(\geqslant 4\)	5	2.221	8.972443

\(\upsilon = 4 - 1 - 1 = 2\)

B1ft

CR : \(\chi^2_2(0.05) \gt 5.991\)

\(\displaystyle\sum\frac{(\mathrm{O} - \mathrm{E})^2}{\mathrm{E}} = \frac{(40 - 34.9937)^2}{34.9937} + \ldots + \frac{(13 - 8.972443)^2}{8.972443}\)
[\(= 0.7161\ldots + 0.3813\ldots + 1.4508\ldots + 1.80789\ldots\)]

\(= 4.356.\) (ans in range 4.2 – 4.4)

Not in critical region
Number of goals scored can follow a Poisson distribution / managers claim is justified

A1 ft

(7)

(12 marks)

Notes

1st B1 Must have both hypotheses and mention Poisson at least once
inclusion of their value for mean in hypotheses is B0 but condone in conclusion

1st M1 for an attempt to pool \(\geqslant 4\)

2nd B1ft for \(n - 1 - 1 = 2\) i.e realising that they must subtract 2 from their \(n\)

3rd B1 for 5.991 only

2nd M1 for an attempt at the test statistic, at least 2 correct expressions/values (to 3sf)

1st A1 for answers in the range 4.2~4.4

2nd A1 for correct comment in context based on their test statistic and their cv that mentions goals or manager. Dependent on 2^nd M1
Condone mention of Po(1.05) in conclusion
Score A0 for inconsistencies e.g. “significant” followed by “manager’s claim is justified”