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.

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

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

Find $w$ .

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.

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.

(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)

(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.

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

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

Expected number of large size eggs

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

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).