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418 lines
14 KiB
HTML
418 lines
14 KiB
HTML
<!doctype html>
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<!-- Mirrored from www.mathsisfun.com/activity/estimation.html by HTTrack Website Copier/3.x [XR&CO'2014], Sat, 29 Oct 2022 00:57:57 GMT -->
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<meta http-equiv="content-type" content="text/html; charset=utf-8">
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<title>Activity: Estimation</title>
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<article id="content" role="main">
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<!-- #BeginEditable "Body" -->
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<h1 class="center">Activity: Estimation</h1>
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<p>This
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activity is about <a
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href="../numbers/estimation.html">estimation</a> and doing a <a href="../definitions/survey.html">survey</a>.</p>
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<h2>How
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many red marbles?</h2>
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<p style="float:left; margin: 0 10px 5px 0;"><img src="images/marbles-asstd.gif" width="99" height="114" alt="marbles assorted" /></p>
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<p>For this experiment you will need a bag of marbles of various
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colors - red, blue, green, yellow etc. The total number of marbles is
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not really important, but you will find it easier to work with a number
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like 50. What is important is that you do not know how many of each
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color there are. So you will have to ask a friend or relative to choose
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the marbles for you.</p>
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<p>You will <a
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href="../numbers/estimation.html">estimate</a> how many red marbles there are in the bag.</p>
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<h3>So why don't you just count them?</h3>
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<p>Of course you could, but this experiment will teach you how to make an
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estimate.</p>
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<p>Scientists and statisticians use this method all the time to
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make estimates in
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real world situations where they can't just simply count.</p>
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<div class="example">
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<h3>Example: Blood Types</h3>
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<p>There are four different human blood types, called A, B, AB and O.</p>
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<p><b>To test everyone, as you can imagine, would be very hard!</b></p>
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<p>So to find out
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how many people in the world have each blood type, scientists take a
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sample of (say) 1,000 people and test their blood type.</p>
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<p>From the sample
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they can estimate how many of each type there are in the world. </p>
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</div>
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<h2>Now you're ready to begin the experiment. It's very simple.</h2>
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<ul>
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<li>Shake
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the marbles in the bag.</li>
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<li>Take
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one marble without looking and record its color.</li>
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<li>Return
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the marble to the bag.</li>
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<li>Repeat
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this process 100 times.</li>
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</ul>
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<h2>Why do you return the marble to the bag?<br></h2>
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<p>So that the conditions for choosing the second
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marble are exactly the same as the conditions for choosing the first
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marble.</p>
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<p>If the first marble you choose is red and you don't replace
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it, then for your second
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choice there is one less red marble in the bag, which makes the chance of choosing another red marble
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less.</p>
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<h2>How do you record the color?</h2>
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<p><strong>You can use a <a href="../data/tally-marks.html">tally/frequency</a> table like this:</strong></p>
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<table border="1" align="center" cellpadding="2"
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cellspacing="2">
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<tbody>
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<tr>
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<td><strong>Color</strong></td>
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<td><strong>Tally</strong></td>
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<td><strong>Frequency</strong></td>
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</tr>
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<tr>
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<td>Red</td>
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<td></td>
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<td></td>
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</tr>
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<tr>
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<td>Blue</td>
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<td></td>
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<td></td>
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</tr>
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<tr>
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<td>Green</td>
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<td></td>
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<td></td>
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</tr>
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<tr>
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<td>Yellow</td>
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<td></td>
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<td></td>
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</tr>
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<tr>
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<td>Purple</td>
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<td></td>
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<td></td>
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</tr>
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<tr>
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<td></td>
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<td></td>
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<td>Total = 100</td>
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</tr>
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</tbody>
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</table>
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<br>
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<br>
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<div class="center80">
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<p>I have included five colors in my table, but yours could be different.</p>
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<p>Once you've completed choosing 100 times, you can work out the <a
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href="../data/relative-frequency.html">relative
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frequency</a> of red by dividing the number of red marbles by 100.</p>
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<p> </p>
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<p>For example, if you record
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<strong>22 red marbles</strong> in your table, then the
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relative frequency of <strong>red</strong>
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is
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<strong>22/100 = 0.22</strong></p>
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</div>
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<h2>How does this help you estimate the
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number of red marbles in the bag?</h2>
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<p>That's easy. If there are <strong>50</strong> marbles in the bag and <strong>red</strong> occurred <strong>22</strong> times out of <strong>100</strong>, then it should occur <strong>11</strong> times out of <strong>50</strong>.</p>
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<p>In other words, the fractions <strong>22/100</strong> and <strong>11/50</strong> are <a
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href="../equivalent_fractions.html">equivalent
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fractions</a>.<br>
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An easier way is just to multiply <strong>0.22 by 50</strong>: <strong>0.22 × 50 = 11</strong></p>
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<p>When you've finished the calculations, you can check how many red
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marbles there really are in the bag.</p>
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<ul>
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<li><strong>How good was your estimate?</strong></li>
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<li><strong>How can you get a better estimate?</strong></li>
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</ul>
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<p>To get a better estimate, don't count straight away.<br>
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Instead, repeat the
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experiment several times and calculate the <a
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href="../mean.html">mean</a> number of red
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marbles.</p>
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<p>Then you can compare your mean number with the actual number in the
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bag.<br>
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You should get a much better estimate, and may even get it exactly
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right.</p>
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<div class="center80">
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<p>As a variation on this experiment, you could use Smarties or M and M's.
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But you may be tempted
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to eat some and ruin the results of your experiment!</p>
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</div>
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<h2>How many A's?</h2>
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<p>Another variation on
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the above experiment can be done using the letters from the game of
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Scrabble. In the game of Scrabble there are 100 tiles. 98 of the tiles
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are inscribed with letters of the alphabet, the other two are blank. Remove the blank ones, so you will have 98 tiles.</p>
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<div class="center80">
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<p>If you don't have the game of Scrabble, then you could make your own
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tiles. You<strong> will need 98 square pieces of cardboard, all the same size:
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2 cm by 2 cm will do. </strong></p>
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<p>Write letters on the tiles as follows:</p>
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<table border="0" align="center">
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<tr>
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<td width="200"><ul>
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<li>9 A's</li>
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<li>2 B's</li>
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<li>2 C's,</li>
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<li>4 D's,</li>
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<li>12
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E's,</li>
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<li>2 F's,</li>
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<li>3 G's,</li>
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<li>2 H's,</li>
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<li>9 I's,</li>
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<li>1 J,</li>
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<li>1 K,</li>
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<li>4 L's,</li>
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<li>2 M's,</li>
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</ul></td>
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<td width="200"><ul>
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<li>6 N's,</li>
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<li>8 O's,</li>
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<li>2 P's,</li>
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<li>1 Q,</li>
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<li>6 R's,</li>
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<li>4 S's,</li>
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<li>6 T's,</li>
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<li>4 U's,</li>
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<li>2 V's,</li>
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<li>2 W's,</li>
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<li>1 X,</li>
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<li>2 Y's</li>
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<li>1
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Z</li>
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</ul></td>
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</tr>
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</table>
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</div>
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<p>Pretend that you don't know
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how many A's there are.
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Now do the experiment in the same way that you did the marbles
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experiment.</p>
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<p><br>
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<strong>Record your results in a table as before:</strong></p>
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<table border="1" align="center" cellpadding="2"
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cellspacing="2">
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<tbody>
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<tr>
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<td><strong>Letter</strong></td>
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<td><strong>Tally</strong></td>
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<td><strong>Frequency</strong></td>
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</tr>
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<tr>
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<td>A</td>
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<td></td>
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<td></td>
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</tr>
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<tr>
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<td>B</td>
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<td></td>
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<td></td>
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</tr>
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<tr>
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<td>C</td>
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<td></td>
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<td></td>
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</tr>
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<tr>
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<td>D</td>
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<td></td>
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<td></td>
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</tr>
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<tr>
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<td>E</td>
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<td></td>
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<td></td>
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</tr>
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<tr>
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<td>etc ...</td>
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<td></td>
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<td></td>
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</tr>
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<tr>
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<td></td>
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<td></td>
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<td>Total = 100</td>
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</tr>
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</tbody>
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</table>
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<div class="center80">
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<p>The only
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difference is that you will need a much longer table going all the way
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to Z.</p>
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<p>When you've
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finished, calculate the <a
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href="../data/relative-frequency.html">relative
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frequency</a> of the letter <strong>A</strong> and
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estimate the number of A's in the bag by multiplying the relative
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frequency by <strong>98</strong>.</p>
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<p>Do this several times and calculate the <a
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href="../mean.html">mean</a>.</p>
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<br>
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</div>
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<h3>What result did you get?</h3>
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<h2>How many people have blood type O?</h2>
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<p>Now you've learned how to make an estimate from a sample,
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you are ready to do a real-life experiment.<br>
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Before you begin you should
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read the pages <a
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href="../data/survey-conducting.html">How to Do
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a Survey</a> and <a href="../data/survey-questionnaire.html">Survey
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Questions</a>.</p>
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<h2><br>
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How many people in the world have
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blood type O?</h2>
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<p>There
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are four different human blood types, called <strong>A</strong>, <strong>B</strong>, <strong>AB</strong> and <strong>O</strong>
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. It is
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obviously impractical to test everybody in the world; so, to find out
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how many people in the world have each blood type, you get
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a
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sample of 100 people and find out their blood type. From the
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sample you can estimate how many people in the world have
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blood type O.</p>
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<p>But this could be a difficult experiment as many people don't even
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know their own blood type!</p>
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<p>And, since you can't test them for blood
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type, you will have to just count the ones who do know. So to get 100 who <b>do</b> know their blood type means a sample size much greater than 100.</p>
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<div class="center80">
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<p><strong>Note:</strong> Depending on where you live in the world, you could use a
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different physical characteristic such as hair color or eye color. But,
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if you live in some countries, you may not have a good variation of
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different colors. That's why I chose blood type.</p>
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</div>
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<h2>Which sample?</h2>
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<p>If you are still at school, then you could use your school population
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do do your survey. Ask permission first, then go around different
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classrooms.</p>
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<p>But you might get a better result doing your survey in a
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local shopping center or mall. When you stop people, tell them that you
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are doing a survey and politely ask them if they would mind telling you
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their blood type.</p>
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<h3>Ignore the ones who refuse to answer or who don't know.</h3>
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<p><strong>Keep going
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until you have 100 positive answers.</strong><br>
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Record your results in a table, as follows:</p>
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<table border="1" align="center" cellpadding="2"
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cellspacing="2">
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<tbody>
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<tr>
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<td><strong>Blood type</strong></td>
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<td><strong>Tally</strong></td>
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<td><strong>Frequency</strong></td>
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</tr>
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<tr>
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<td>A</td>
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<td></td>
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<td></td>
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</tr>
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<tr>
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<td>B</td>
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<td></td>
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<td></td>
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</tr>
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<tr>
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<td>AB</td>
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<td></td>
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<td></td>
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</tr>
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<tr>
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<td>O</td>
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<td></td>
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<td></td>
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</tr>
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<tr>
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<td></td>
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<td></td>
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<td>Total = 100</td>
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</tr>
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</tbody>
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</table>
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<br>
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When you have
|
|
finished, you can work out the relative frequency of blood type <strong>O</strong> from
|
|
your sample.<br>
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|
Divide the number with blood type <strong>O </strong>by <strong>100</strong>.
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|
<p>You can expect to get a better result if you use a bigger sample, or if
|
|
you take several samples and calculate the mean.</p>
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|
<p>Now all you need to know is how many people there are in the world. A quick search on the internet should answer this for you.</p>
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<h2>Can you now estimate how many people in the world have blood type O?</h2>
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<p> </p>
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<p> </p>
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