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Date May Example question Marks available 8 Reference code EXM.1.AHL.TZ0.55
Level Additional Higher Level Paper Paper 1 Time zone Time zone 0
Command term Test and Calculate 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 χ 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  = 1 × 29 + + 6 × 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.

p ^ = 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         ( f 0 f e ) 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 ( f 0 f e ) 2 values, (A1) for combining cells.

χ 2 = 0.910 9.046 + + 5.345 9.688 = 1.56       (A1)

OR

χ 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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Topic 4—Statistics and probability

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