<p>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.</p>
<div class="content-block content-block--table"><div class="table-container"><table style="table-layout: fixed; width: 760px;"><colgroup><col style="width: 12.5000%;"><col style="width: 12.5000%;"><col style="width: 12.5000%;"><col style="width: 12.5000%;"><col style="width: 12.5000%;"><col style="width: 12.5000%;"><col style="width: 12.5000%;"><col style="width: 12.5000%;"></colgroup><thead><tr><th style="text-align: center; --cell-justify: center"><strong>Length $l$ cm</strong></th><th style="text-align: center; --cell-justify: center">$0 \leq l \leq 10$</th><th style="text-align: center; --cell-justify: center">$10 \leq l \leq 20$</th><th style="text-align: center; --cell-justify: center">$20 \leq l \leq 30$</th><th style="text-align: center; --cell-justify: center">$30 \leq l \leq 40$</th><th style="text-align: center; --cell-justify: center">$40 \leq l \leq 60$</th><th style="text-align: center; --cell-justify: center">$60 \leq l \leq 75$</th><th style="text-align: center; --cell-justify: center">$75 \leq l \leq 100$</th></tr></thead><tbody><tr><td style="text-align: center; --cell-justify: center"><strong>Frequency</strong></td><td style="text-align: center; --cell-justify: center">$30$</td><td style="text-align: center; --cell-justify: center">$40$</td><td style="text-align: center; --cell-justify: center">$50$</td><td style="text-align: center; --cell-justify: center">$30$</td><td style="text-align: center; --cell-justify: center">$33$</td><td style="text-align: center; --cell-justify: center">$11$</td><td style="text-align: center; --cell-justify: center">$6$</td></tr></tbody></table></div></div>
<ol class="abc-list"><li data-list-item-id="item-1786952659210-6pswlu03l" data-list-item-path="item-1786952659210-6pswlu03l"><div class="list-item-paragraph"><p>Calculate an estimate for the standard deviation of the lengths of the fish.</p><span class="inline-marks">[3]</span></div></li><li data-list-item-id="item-1786952668666-iv86d82bf" data-list-item-path="item-1786952668666-iv86d82bf"><div class="list-item-paragraph"><p>A cumulative frequency diagram is given below for the lengths of the fish.</p></div><div class="content-block list-item-image" style="text-align: center;"><img src="https://vocwkpgapvksartxhhor.supabase.co/storage/v1/object/sign/question-images/pasted/1786952729824-402ocbvglv9.png?token=eyJraWQiOiJiZTM3ZjNhZS05OWM4LTQ5ZGQtYjRiNS1iMTBhYmJhNjY5MWYiLCJhbGciOiJIUzI1NiJ9.eyJ1cmwiOiJxdWVzdGlvbi1pbWFnZXMvcGFzdGVkLzE3ODY5NTI3Mjk4MjQtNDAyb2NidmdsdjkucG5nIiwic2NvcGUiOiJkb3dubG9hZCIsImlhdCI6MTc5MDg2MjE4MSwiZXhwIjoxNzkwODYyNDgxfQ.Vk_x4t_rGQfEw1UzJ2I_qfw13Lfc-jlxZ-C1C0C9s6g" alt="List item image" class="aligncenter custom-width" style="max-width: min(399px, 100%); height: auto;" /></div><div class="list-item-paragraph"><p>Use the graph to answer the following.</p></div><ol class="i-list"><li data-list-item-id="subitem-1786952788172-y5134hiqj" data-list-item-path="item-1786952668666-iv86d82bf/subitem-1786952788172-y5134hiqj"><div class="list-item-paragraph"><p>Estimate the interquartile range.</p></div></li><li data-list-item-id="subitem-1786952789148-m6cxyacyq" data-list-item-path="item-1786952668666-iv86d82bf/subitem-1786952789148-m6cxyacyq"><div class="list-item-paragraph"><p>Given that $40\%$ of the fish have a length more than $k$ cm, find the value of $k$.</p><span class="inline-marks">[6]</span></div></li></ol></li></ol>
<p>In order to sell the fish, the fisherman classifies them as small, medium or large.</p>
<p>$\quad$Small fish have a length less than $20$ cm.<br>$\quad$Medium fish have a length greater than or equal to $20$ cm but less than $60$ cm.<br>$\quad$Large fish have a length greater than or equal to $60$ cm.</p>
<ol class="abc-list" start="3"><li data-list-item-id="item-1786952969234-3n3i766c1" data-list-item-path="item-1786952969234-3n3i766c1"><div class="list-item-paragraph"><p>Write down the probability that a fish is small.</p><span class="inline-marks">[2]</span></div></li></ol>
<p>The cost of a small fish is $\$4$, a medium fish $\$10$, and a large fish $\$12$.</p>
<ol class="abc-list" start="4"><li data-list-item-id="item-1786953034235-xft3apnai" data-list-item-path="item-1786953034235-xft3apnai"><div class="list-item-paragraph"><p>Copy and complete the following table, which gives a probability distribution for the cost $\$X$.</p><span class="inline-marks">[2]</span></div><div class="table-container"><table style="table-layout: fixed; width: 300px;"><colgroup><col style="width: 25.0000%;"><col style="width: 25.0000%;"><col style="width: 25.0000%;"><col style="width: 25.0000%;"></colgroup><thead><tr><th style="text-align: center; --cell-justify: center"><strong>Cost $\$X$</strong></th><th style="text-align: center; --cell-justify: center">$4$</th><th style="text-align: center; --cell-justify: center">$10$</th><th style="text-align: center; --cell-justify: center">$12$</th></tr></thead><tbody><tr><td style="text-align: center; --cell-justify: center">$P(X = x)$</td><td style="text-align: center; --cell-justify: center"></td><td style="text-align: center; --cell-justify: center">$0.565$</td><td style="text-align: center; --cell-justify: center"></td></tr></tbody></table></div></li><li data-list-item-id="item-1786953163782-t0dtd1t6i" data-list-item-path="item-1786953163782-t0dtd1t6i"><div class="list-item-paragraph"><p>Find E(X).</p><span class="inline-marks">[2]</span></div></li></ol>

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.

<p>Calculate an estimate for the standard deviation of the lengths of the fish.</p>

<p>A cumulative frequency diagram is given below for the lengths of the fish.</p>

<p>Use the graph to answer the following.</p>

In order to sell the fish, the fisherman classifies them as small, medium or large.

$\quad$Small fish have a length less than $20$ cm.
$\quad$Medium fish have a length greater than or equal to $20$ cm but less than $60$ cm.
$\quad$Large fish have a length greater than or equal to $60$ cm.

<p>Write down the probability that a fish is small.</p>

The cost of a small fish is $\$4$, a medium fish $\$10$, and a large fish $\$12$.

<p>Copy and complete the following table, which gives a probability distribution for the cost $\$X$.</p>

Statistics

International Baccalaureate Mathematics: Applications and Interpretation Standard Level