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Date	May Example question	Marks available	1	Reference code	EXM.1.AHL.TZ0.55
Level	Additional Higher Level	Paper	Paper 1	Time zone	Time zone 0
Command term	Hence and Estimate	Question number	55	Adapted from	N/A

Question

Eggs at a farm are sold in boxes of six. Each egg is either brown or white. The owner believes that the number of brown eggs in a box can be modelled by a binomial distribution. He examines 100 boxes and obtains the following data.

Calculate the mean number of brown eggs in a box.

[1]

a.i.

Hence estimate $p$ , the probability that a randomly chosen egg is brown.

[1]

a.ii.

By calculating an appropriate ${\chi ^2}$ statistic, test, at the 5% significance level, whether or not the binomial distribution gives a good fit to these data.

[8]

b.

Markscheme

Note: Candidates may obtain slightly different numerical answers depending on the calculator and approach used. Use discretion in marking.

Mean $= \frac{{1 \times 29 + \ldots + 6 \times 1}}{{100}} = 1.98$ (A1)

[1 mark]

a.i.

Note: Candidates may obtain slightly different numerical answers depending on the calculator and approach used. Use discretion in marking.

$\widehat p = \frac{{1.98}}{6} = 0.33$ (A1)

[1 mark]

a.ii.

Note: Candidates may obtain slightly different numerical answers depending on the calculator and approach used. Use discretion in marking.

The calculated values are

${f_0}$ ${f_e}$ ${\left( {{f_0} - {f_e}} \right)^2}$
10 9.046 0.910
29 26.732 5.14 (M1)
31 32.917 3.675   (A1)
18 21.617 13.083   (A1)
12 9.688 5.345   (A1)

Note: Award (M1) for the attempt to calculate expected values, (A1) for correct expected values, (A1) for correct ${\left( {{f_0} - {f_e}} \right)^2}$ values, (A1) for combining cells.

${\chi ^2} = \frac{{0.910}}{{9.046}} + \ldots + \frac{{5.345}}{{9.688}} = 1.56$ (A1)

OR

${\chi ^2} = 1.56$ (G5)

Degrees of freedom = 3; Critical value = 7.815

(or p-value = 0.668 (or 0.669)) (A1)(A1)

We conclude that the binomial distribution does provide a good fit. (R1)

[8 marks]

b.

Examiners report

[N/A]

a.i.

[N/A]

a.ii.

[N/A]

b.

Syllabus sections

Topic 4—Statistics and probability » SL 4.8—Binomial distribution

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