AS June 2019 Q1

EdexcelAS paperCurrent spec6 marksIncludes hypothesis testingChi-Squared Tests

1. A leisure club offers a choice of one of three activities to its 150 members on a Tuesday evening. The manager believes that there may be an association between the choice of activity and the age of the member and collected the following data.

Activity Age \(a\) years	Badminton	Bowls	Snooker
\(a \lt 20\)	9	3	3
\(20 \leqslant a \lt 40\)	10	10	14
\(40 \leqslant a \lt 50\)	16	15	5
\(50 \leqslant a \lt 60\)	15	13	11
\(a \geqslant 60\)	4	19	3

(a) Write down suitable hypotheses for a test of the manager’s belief. (1)

The manager calculated expected frequencies to use in the test.

(b) Calculate the expected frequency of members aged 60 or over who choose snooker, used by the manager. (1)

The test statistic used to test the manager’s belief is 19.583

(d) Using a 5% level of significance, complete the test of the manager’s belief. (2)

Mark scheme (a)

Scheme	Marks	AO
\(\mathrm{H}_0\): There is no association between age and activity \(\mathrm{H}_1\): There is an association between age and activity	B1	1.2
	(1)

Notes

B1 for both hypotheses in terms of “association” or “independence”. Must mention age and activity (or sport). [Use of “relationship” or “link” here is B0 but allow for last B1ft]

Scheme	Marks	AO
\(\dfrac{26 \times 36}{150} = 6.24\)	B1	1.1b
	(1)

Mark scheme (c)

Scheme	Marks	AO
Since expected value in \(a \lt 20\) and snooker \(= 3.6 \lt 5\) we amalgamate two rows	B1	2.4
Table is now \(4 \times 3\) so degrees of freedom is \((4-1) \times (3-1) = 6\) (*)	B1*	1.1b
	(2)

Notes

1^st B1 for a reason to get a 4 x 3 table based on amalgamation of rows
Must mention \(a \lt 20\) and snooker and see 3.6 and be combining rows (not columns)

2^nd B1* for 6 degrees of freedom clearly coming from a 4 x 3 table formed from amalgamation of rows. [ 8 – 2 = 6 is B0 ]

Mark scheme (d)

Scheme	Marks	AO
Critical value \(\chi^2_6(5\%) = 12.592\)	B1	1.1b
[Significant result]: so there is evidence to support manager’s belief	B1ft	2.2b
	(2)
	(6 marks)

Notes

1^st B1 for correct critical value (allow 12.6 or 12.59 or awrt 12.592)
NB \(p\)-value \(= 0.0032839\ldots\) so allow awrt 0.00328

2^nd B1ft for a correct comparison and conclusion (ft their cv) [Independent of hypotheses]
e.g. there is an “association” or “relationship” or “link” between age and activity is OK
BUT there is a “correlation” between age and activity is B0

Do not accept contradictory contextual statements e.g. “manager’s belief supported , there is no association between age and activity”