AS June 2023 Q2

EdexcelAS paperCurrent spec6 marksIncludes hypothesis testingChi-Squared Tests

2. A bag contains a large number of balls, all of the same size and weight. The balls are coloured Red, Blue or Yellow.

Jasmine asks each child in a group of 150 children to close their eyes, select a ball from the bag and show it to her. The child then replaces the ball and repeats the process a second time.

If both balls are the same colour the child receives a prize.

The results are given in the table below.

1st colour → 2nd colour ↓	Red	Blue	Yellow	Total
Red	31	11	18	60
Blue	8	10	9	27
Yellow	21	9	33	63
Total	60	30	60	150

Jasmine carries out a test, at the 5% level of significance, to see whether or not the colour of the 2nd ball is independent of the colour of the 1st ball.

(a) Calculate the expected frequencies for the cases where both balls are the same colour. (2)

The test statistic Jasmine obtained was 12.712 to three decimal places.

(b) Use this value to complete the test, stating the critical value and conclusion clearly. (3)

With reference to your calculations in part (a) and the nature of the experiment,

Mark scheme (a)

Scheme	Marks	AO
\(\mathrm{E(RR)} = \dfrac{60 \times 60}{150}\) or \(\mathrm{E(BB)} = \dfrac{30 \times 27}{150}\) or \(\mathrm{E(YY)} = \dfrac{60 \times 63}{150}\)	M1	1.1b
\(\mathrm{E(RR)} = \underline{\mathbf{24}}\) and \(\mathrm{E(BB)} = \underline{\mathbf{5.4}}\) and \(\mathrm{E(YY)} = \underline{\mathbf{25.2}}\)	A1	1.1b
	(2)

Notes

M1 for at least one correct calculation seen

A1 for all 3 expected frequencies correct, must be exact.

Mark scheme (b)

Scheme	Marks	AO
[\(\mathrm{E(BB)} \gt 5\) so no need for pooling] \(\nu = (3-1)(3-1) = 4\)	B1	3.1b
Critical value: \(\chi_4^2(5\%) = \underline{\mathbf{9.488}}\)	B1ft	1.1b
(significant) evidence that the colour of balls is not independent	B1ft	2.2b
	(3)

Notes

1^st B1 for 4 degrees of freedom

2^nd B1ft for awrt 9.488 or ft their degrees of freedom for 5% cv

3^rd B1ft for correct contextual conclusion mentioning “colour” and “independent”
Must be consistent with their cv
Allow associated instead of not independent

Mark scheme (c)

Scheme	Marks	AO
\(O_i \gt E_i\) for pairings of the same colour and some children may be cheating when selecting the 2^nd ball so choice not independent (o.e.)	B1	3.5a
	(1)
	(6)

Notes

B1 Stating or indicating that \(O_i \gt E_i\) and a suitable reason for the lack of independence that would invalidate the modelling assumption. Such as:

cheating/looking
The first ball being replaced by the child so more likely to be picked again