October 2021 Paper 2 Q13

OCR MEICurrent spec7 marksBinomial Distribution Data Processing

13 At a certain factory Christmas tree decorations are packed in boxes of 10.

The quality control manager collects a random sample of 100 boxes of decorations and records the number of decorations in each box which are damaged.

His results are displayed in Fig. 13.1.

Number of damaged decorations	0	1	2	3	4	5 or more
Number of boxes	19	35	28	13	5	0

Fig. 13.1

(a) Calculate

the mean number of damaged decorations per box,
the standard deviation of the number of damaged decorations per box.

[2]

It is believed that the number of damaged decorations in a box of 10, \(X\), may be modelled by a binomial distribution such that \(X \sim \mathrm{B}(n, p)\).

(b) State suitable values for \(n\) and \(p\). [1]

(c) Use the binomial model to complete the copy of Fig. 13.2 in the Printed Answer Booklet, giving your answers correct to 1 decimal place. [3]

Number of damaged decorations	0	1	2	3	4	5 or more
Observed number of boxes	19	35	28	13	5	0
Expected number of boxes

Fig. 13.2

(d) Explain whether the model is a good fit for these data. [1]

Scheme	Marks	AO
mean = 1.5 BC	B1	1.1
sd = 1.1, 1.10 or awrt 1.096 BC	B1	1.1
	[2]

Scheme	Marks	AO
\(n = 10\) and \(p = 0.15\)	B1	3.3
	[1]

Mark scheme (c)

Scheme

Marks

N	0	1	2	3	4	5 +
O	19	35	28	13	5	0
E	19.7	34.7	27.6	13.0	4.0	1.0

M1
M1
A1

3.4
3.4
1.1

[3]

Notes

M1: use of \(\mathrm{B}(10, p)\) FT their \(p\) to find at least 2 probabilities; may be unsimplified
must see at least 2 probabilities for M1

M1: multiplying their binomial probabilities by 100
must see at least 2 expected values calculated for M1

A1: all correct to 1 dp or better; must add up to 100

Mark scheme (d)

Scheme	Marks	AO
close match between theoretical (or expected) and observed frequencies so model is a good fit	B1	3.5a
	[1]

Notes

B1: FT dependent on award of M1M1 in (c) and all frequencies calculated