A fisherman catches 200 fish to sell. He measures the lengths, $l$ cm, of these fish, and the results are shown in the frequency table below. Length $l$ cm $0\le l<10$ $10\le l<20$ $20\le l<30$ $30\le l<40$ $40\le l<60$ $60\le l<75$ $75\le l<100$ Frequency 30 40 50 30 33 11 6 Calculate an estimate for the standard deviation of the lengths of the fish. A cumulative frequency diagram is given below for the lengths of the fish. Use the graph to answer the following. Estimate the interquartile range. Given that 40 % of the fish have a length more than k cm, find the value of k. In order to sell the fish, the fisherman classifies them as small, medium or large. Small fish have a length less than 20 cm. Medium fish have a length greater than or equal to 20 cm but less than 60 cm. Large fish have a length greater than or equal to 60 cm. Write down the probability that a fish is small. The cost of a small fish is $4, a medium fish $10, and a large fish $12. Copy and complete the following table, which gives a probability distribution for the cost $X$. Find $E(X)$.

Analyse grouped lengths with standard deviation, cumulative frequency, quartiles and an expected-cost distribution.

15 marks

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A fisherman catches 200 fish to sell. He measures the lengths, $l$ cm, of these fish, and the results are shown in the frequency table below.

Length $l$ cm	$0\le l<10$	$10\le l<20$	$20\le l<30$	$30\le l<40$	$40\le l<60$	$60\le l<75$	$75\le l<100$
Frequency	30	40	50	30	33	11	6

Calculate an estimate for the standard deviation of the lengths of the fish.[3]
A cumulative frequency diagram is given below for the lengths of the fish. Use the graph to answer the following.[6]
1. Estimate the interquartile range.
2. Given that 40 % of the fish have a length more than k cm, find the value of k. In order to sell the fish, the fisherman classifies them as small, medium or large. Small fish have a length less than 20 cm. Medium fish have a length greater than or equal to 20 cm but less than 60 cm. Large fish have a length greater than or equal to 60 cm.
Write down the probability that a fish is small. The cost of a small fish is $4, a medium fish $10, and a large fish $12.[2]
Copy and complete the following table, which gives a probability distribution for the cost $X$.[2]
Cost $X$ 4 10 12
$P(X=x)$ 0.565
Find $E(X)$.[2]

Cost \(X\)	4	10	12
\(P(X=x)\)		0.565

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Answer:$P(\text{small})=\frac{70}{200}=0.35$. The cost distribution is $P(4)=0.35$, $P(10)=0.565$, $P(12)=0.085$, so $E(X)=8.07$.

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