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Date May Specimen Marks available 3 Reference code SPM.2.sl.TZ0.3
Level SL only Paper 2 Time zone TZ0
Command term Estimate Question number 3 Adapted from N/A

Question

The Brahma chicken produces eggs with weights in grams that are normally distributed about a mean of \(55{\text{ g}}\) with a standard deviation of \(7{\text{ g}}\). The eggs are classified as small, medium, large or extra large according to their weight, as shown in the table below.

Sketch a diagram of the distribution of the weight of Brahma chicken eggs. On your diagram, show clearly the boundaries for the classification of the eggs.

[3]
a.

An egg is chosen at random. Find the probability that the egg is
(i)     medium;
(ii)    extra large.

[4]
b.

There is a probability of \(0.3\) that a randomly chosen egg weighs more than \(w\) grams.

Find \(w\) .

[2]
c.

The probability that a Brahma chicken produces a large size egg is \(0.121\). Frank’s Brahma chickens produce \(2000\) eggs each month.

Calculate an estimate of the number of large size eggs produced by Frank’s chickens each month.

[2]
d.

The selling price, in US dollars (USD), of each size is shown in the table below.

The probability that a Brahma chicken produces a small size egg is \(0.388\).

Estimate the monthly income, in USD, earned by selling the \(2000\) eggs. Give your answer correct to two decimal places.

[3]
e.

Markscheme

(A1) for normal curve with mean of \(55\) indicated
(A1) for three lines in approximately the correct position
(A1) for labels on the three lines     (A1)(A1)(A1)

a.

(i)     \({\text{P}}(53 \leqslant {\text{Weight}} < 63) = 0.486\) (\(0.485902 \ldots \))     (M1)(A1)(G2)


Note: Award (M1) for correct region indicated on labelled diagram.


(ii)    \({\text{P}}({\text{Weight}} > 73) = 0.00506\) (\(0.00506402\))     (M1)(A1)(G2)


Note: Award (M1) for correct region indicated on labelled diagram.

b.

\({\text{P}}({\text{Weight}} > w) = 0.3\)     (M1)
\(w = 58.7\) (\(58.6708 \ldots \))     (A1)(G2)


Note: Award (M1) for correct region indicated on labelled diagram.

c.

Expected number of large size eggs

\( = 2000(0.121)\)     (M1)
\( = 242\)     (A1)(G2)

d.

Expected income
\( = 2000 \times 0.30 \times 0.388 + 2000 \times 0.50 \times 0.486 + 2000 \times 0.65 \times 0.121 + 2000 \times 0.80 \times 0.00506\)     (M1)(M1)

Note: Award (M1) for their correct products, (M1) for addition of 4 terms.

 

\( = 884.20{\text{ USD}}\)     (A1)(ft)(G3)

Note: Follow through from part (b).

e.

Examiners report

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b.
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d.
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e.

Syllabus sections

Topic 1 - Number and algebra » 1.2 » Approximation: decimal places, significant figures.
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