June 2025 Paper 2 Q11

OCR MEICurrent spec7 marksBinomial Distribution Data Processing

11 Apples are sold in packets of 4. Ling and Sam are investigating the frequency, \(f\), of the number of bruised apples in a packet, \(x\). They each collect a random sample of packets, note the number of bruised apples in each packet and draw a diagram to represent their data.

Ling’s results are shown in Table 11.1 and Fig. 11.1.

Table 11.1

\(x\)	0	1	2	3	4
\(f\)	19	2	0	2	2

Fig. 11.1: bar chart of frequency against number of bruised apples for Ling: 19, 2, 0, 2, 2 for x = 0 to 4 — Fig. 11.1

Sam’s results are shown in Table 11.2 and Fig. 11.2.

Table 11.2

\(x\)	0	1	2	3	4
\(f\)	39	1	0	3	7

Fig. 11.2: bar chart of frequency against number of bruised apples for Sam: 39, 1, 0, 3, 7 for x = 0 to 4 — Fig. 11.2

(a) Explain why Sam’s diagram is likely to be a better representation of the true distribution of the number of bruised apples in a packet than Ling’s diagram. [1]

(b) Calculate the mean number of bruised apples per packet for Sam’s data. [1]

Sam thinks that the distribution of bruised apples may be modelled by a binomial distribution.

(c) Use your answer to part (b) to calculate the value of \(p\), the probability that an apple selected at random is bruised, for Sam’s model. [1]

(d) Calculate the theoretical frequency distribution of the number of bruised apples per packet for Sam’s model, giving your answers correct to 2 decimal places. [3]

(e) Comment on whether Sam’s model appears to be a good fit for the data. [1]

Scheme	Marks	AO
because Sam used a larger sample oe	B1	2.4
	[1]

Scheme	Marks	AO
0.76 or \(\frac{38}{50}\) oe BC	B1	1.1
	[1]

Scheme	Marks	AO
0.19	B1	3.3
	[1]

Mark scheme (d)

Scheme

Marks

\(x\)	0	1	2	3	4
\(f\)	21.52	20.19	7.11	1.11	0.07

B1
B1
B1

3.4
3.4
1.1

[3]

Notes

B1: for 1 correct frequency to 2 d.p.

B1: for 3 correct frequencies to 2 d.p.

B1: for all 5 correct to 2 d.p.

FT use of B(4, their 0.19);
allow SC2 for all 5 frequencies awrt 22, 20, 7, 1, 0 or better
allow SC1 for 3 of the above frequencies awrt nearest whole number or better
if B0B0B0 allow SC1 for probabilities: 0.430467…, 0.403895…, 0.142111…, 0.022223…, 0.001303 given to 3 dp or more; FT use of B(4, their 0.19) with \(0 \lt p \lt 1\)

Mark scheme (e)

Scheme	Marks	AO
by comparing theoretical and observed frequencies (or probabilities), the model is a (very) poor fit oe	B1	3.5a
	[1]

Notes

B1: allow comparison of one pair of frequencies or one pair of probabilities;