A2 June 2022 Q1

EdexcelCurrent spec9 marksIncludes hypothesis testingChi-Squared Tests

1. A researcher is investigating the number of female cubs present in litters of size 4
He believes that the number of female cubs in a litter can be modelled by \(\mathrm{B}(4, 0.5)\)
He randomly selects 100 litters each of size 4 and records the number of female cubs.
The results are recorded in the table below.

Number of female cubs	0	1	2	3	4
Observed number of litters	10	33	33	15	9

He calculated the expected frequencies as follows

Number of female cubs	0	1	2	3	4
Expected number of litters	6.25	\(r\)	\(s\)	\(r\)	6.25

(a) Find the value of \(r\) and the value of \(s\) (3)

(b) Carry out a suitable test, at the 5% level of significance, to determine whether or not the number of female cubs in a litter can be modelled by \(\mathrm{B}(4, 0.5)\)
You should clearly state your hypotheses and the critical value used. (6)

Mark scheme (a)

Scheme	Marks	AO
\(r = \mathrm{P}(X = 3) \times 100\) or \(r = \mathrm{P}(X = 1) \times 100\) or \(s = \mathrm{P}(X = 2) \times 100\)	M1	3.4
\(r = \underline{25}\) (value may be in table)	A1	1.1b
\(s = \underline{37.5}\) (value may be in table)	A1	1.1b
	(3)

Notes

M1 Using the Binomial model to expected value. Allow if both probs 0.25 and 0.375 seen
May be implied by a correct value of \(r\) or \(s\). Alternatives \(r = 6.25 \times 4\) or \(s = 6.25 \times 6\)

1^st A1 for \(r = 25\)

2^nd A1 for \(s = 37.5\)

SC “B1” If M0 scored but their values of \(r\) and \(s\) satisfy \(2r + s = 87.5\) score as M0A0A1

Mark scheme (b)

Scheme

Marks

\(\mathrm{H}_0\): \(\mathrm{B}(4, 0.5)\) is a suitable model (o.e.) Condone \(\mathrm{B}(0.5, 4)\)
\(\mathrm{H}_1\): \(\mathrm{B}(4, 0.5)\) is not a suitable model (o.e.)

2.5

\(\dfrac{(O_i - E_i)^2}{E_i}\)	2.25	2.56	0.54	4	1.21
\(\dfrac{O_i^2}{E_i}\)	16	43.56	29.04	9	12.96

1.1b

\(\displaystyle\sum\frac{(O_i - E_i)^2}{E_i} = 10.56\) or \(\displaystyle\sum\frac{O_i^2}{E_i} - N = 110.56 - 100 = 10.56 \ \left(= \frac{264}{25}\right)\)

1.1b

\(\nu = 5 - 1 = 4\)

1.1b

CV = 9.488 (Calc 9.487729035…)

B1ft

1.1b

Significant so there is evidence that the researcher's model is not suitable

2.2b

(6)

Total 9

Notes

1^st B1 Both hypotheses correct using the correct notation in at least one or written in full e.g. binomial with \(n = 4\) and \(p = 0.5\)

M1 Calculating either \(\dfrac{(O_i - E_i)^2}{E_i}\) or \(\dfrac{O_i^2}{E_i}\) at least 4 correct. Implied by sight of awrt 10.6

1^st A1 Allow 10.6 (from correct working)

2^nd B1 Correct dof May be implied by CV of 9.48 or 9.49 or better

3^rd B1ft For 9.488 or better. Can ft their dof NB \(\chi^2_3(5\%) = 7.815\) (allow awrt 7.815)

2^nd A1 Indep’ of hypotheses but dep on 1^st A1
Evaluating the outcome by drawing a correct inference.
Compatible with comparison of 10.56 or 10.6 with their CV (which must be > 1)
They must say model not suitable (o.e.)
They do not need to state the comparison or say reject \(\mathrm{H}_0\) etc
No need to explicitly see \(\mathrm{B}(4, 0.5)\) mentioned here