S3 June 2012 Q4

EdexcelOld spec10 marksIncludes hypothesis testingChi-Squared Tests

4. Two breeds of chicken are surveyed to measure their egg yield. The results are shown in the table below.

Egg yield Breed	Low	Medium	High
Leghorn	22	52	26
Cornish	14	32	4

Showing each stage of your working clearly, test, at the 5% significance level, whether or not there is an association between egg yield and breed of chicken. State your hypotheses clearly. (10)

Mark scheme

Scheme

Marks

\(\mathrm{H}_0\) : Egg yield and breed of chicken are independent (not associated)
\(\mathrm{H}_1\) : Egg yield and breed of chicken are dependent (associated)

Egg Yield Breed	Low	Medium	High	Total
Leghorn	\(\dfrac{100 \times 36}{150} = 24\)	\(\dfrac{100 \times 84}{150} = 56\)	\(\dfrac{100 \times 30}{150} = 20\)	100
Cornish	\(\dfrac{50 \times 36}{150} = 12\)	\(\dfrac{50 \times 84}{150} = 28\)	\(\dfrac{50 \times 30}{150} = 10\)	50
Total	36	84	30	150

M1A1

\(O\)	\(E\)	\(\displaystyle\sum\frac{(O - E)^2}{E}\)	\(\displaystyle\sum\frac{O^2}{E}\)
22	24	0.166667	20.2
52	56	0.285714	48.3
26	20	1.8	33.8
14	12	0.333333	16.3
32	28	0.571429	36.6
4	10	3.6	1.6

M1A1

\(\displaystyle\sum\frac{(O - E)^2}{E} = 6.757\ldots\) or \(\displaystyle\sum\frac{O^2}{E} - 150 = 6.757\ldots\)

\(\nu = 2, \chi^2_2(5\%) = 5.991\)

B1B1ft

(\(6.757 \gt 5.991\) so sufficient evidence to) reject \(\mathrm{H}_0\)

Egg yield and breed of chicken are dependent (associated)

(10 marks)

Notes

B1 for both hypotheses. Must mention “yield” and “breed” in both but condone ditto marks.
Use of “relationship” or “correlation” or “connection” is B0

1st M1 for some use of \(\dfrac{\text{Row Total} \times \text{Col.Total}}{\text{Grand Total}}\). May be implied by a correct \(E_i\)

1st A1 for all expected frequencies correct

2nd M1 for at least two correct terms or correct expressions with their \(E_i\)

2nd A1 for all correct terms. May be implied by a correct answer (2 sf or better)

3rd M1 for a correct statement linking their test statistic and their cv . Must be \(\chi^2\) not normal.

4th A1 for a correct comment in context - must mention “egg yield” and “breed of chicken” - condone “relationship” or “connection” here but not “correlation”. No follow through e.g. “There is no evidence of a relationship between egg yield and breed of chicken” is A0 whatever their test stat or cv.

(corrected from the printed mark scheme: the alternative calculation is printed as \(\sum\frac{O^2}{E} - 100\); the grand total is 150, so it is \(\sum\frac{O^2}{E} - 150 = 156.757\ldots - 150 = 6.757\ldots\))