S3 June 2007 Q4

EdexcelOld spec13 marksIncludes hypothesis testingChi-Squared Tests

4. A quality control manager regularly samples 20 items from a production line and records the number of defective items \(x\). The results of 100 such samples are given in Table 1 below.

\(x\)	0	1	2	3	4	5	6	7 or more
Frequency	17	31	19	14	9	7	3	0

Table 1

(a) Estimate the proportion of defective items from the production line. (2)

The manager claimed that the number of defective items in a sample of 20 can be modelled by a binomial distribution. He used the answer in part (a) to calculate the expected frequencies given in Table 2.

\(x\)	0	1	2	3	4	5	6	7 or more
Expected frequency	12.2	27.0	\(r\)	19.0	\(s\)	3.2	0.9	0.2

Table 2

(b) Find the value of \(r\) and the value of \(s\) giving your answers to 1 decimal place. (3)

(d) Explain what the analysis in part (c) tells the manager about the occurrence of defective items from this production line. (1)

Mark scheme (a)

Scheme	Marks
\(\dfrac{0 \times 17 + 1 \times 31 + \ldots}{17 + 31 + \ldots} = \left(\dfrac{200}{100} = 2\right)\), \(\hat{p} = \dfrac{2}{20} = \underline{0.1}\) (Accept \(\dfrac{2}{20}\) or 2 per 20)	M1, A1
	(2)

Notes

M1 for attempt to find mean or \(\hat{p}\) (as printed or better). The 0.1 must be seen in part (a).

Mark scheme (b)

Scheme	Marks
e.g. \(r = 100 \times \dbinom{20}{2}(0.1)^2(0.9)^{18}\)	M1
\(r = 28.5\), \(s\) = AWRT 9	A1, A1
	(3)

Notes

M1 for correct expression for \(r\) or \(s\) using the binomial distribution. Follow through their \(\hat{p}\).

Mark scheme (c)

Scheme

Marks

\(x\)	0	1	2	3	\(\geqslant 4\)
\(O_i\)	17	31	19	14	19
\(E_i\)	12.2	27.0	28.5	19.0	13.3
\(\dfrac{(O - E)^2}{E}\)	1.89	0.59	3.17	1.32	2.44

Pooling

\(\displaystyle\sum \frac{(O - E)^2}{E} =\) AWRT 9.4

M1A1c.a.o.

\(\nu = 5 - 2 = 3, \quad \chi^2_3(5\%) = 7.815\)

B1ft, B1ft

\(\mathrm{H}_0\) : Binomial distribution is a good/suitable model/fit [Condone: B(20, 0.1) is…]
\(\mathrm{H}_1\) : Binomial distribution is not a suitable model both

(Significant result) Binomial distribution is not a suitable model

A1cao

(7)

Notes

1^st M1 for some pooling (accept \(x \geqslant 5\), obs.freq. …14, 9, 10 and exp.freq. 19.0, \(s\), 4.3)

2^nd M1 for calculation of test statistic (N.B. \(x \geqslant 5\) gives 14.5). One correct term seen.

1^st B1ft for number of classes – 2 (N.B. \(x \geqslant 5\) will have 6 – 2 = 4)

2^nd B1ft for the appropriate tables value, ft their degrees of freedom. (NB \(\chi^2_4(5\%) = 9.488\))

3^rd B1 (for hypotheses) allow just “\(X \sim \mathrm{B}(20, 0.1)\)” for null etc.

2^nd A1 for correctly rejecting Binomial model. No ft and depends on 2^nd M1.

Mark scheme (d)

Scheme	Marks
defective items do not occur independently or not with constant probability	B1ft
	(1)
	(13 marks)

Notes

B1ft for independence or constant probability – must mention defective items or defectives
Follow through their conclusion in (c). So if they do not reject they may say “defectives occur with probability 0.1”. Stating the value implies constant probability.