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	<title>2657 Productions News &#187; Useless Knowledge</title>
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			<item>
		<title>Sampling with replacement in R</title>
		<link>http://news.mrdwab.com/2009-11-30/sampling-with-replacement-in-r/</link>
		<comments>http://news.mrdwab.com/2009-11-30/sampling-with-replacement-in-r/#comments</comments>
		<pubDate>Mon, 30 Nov 2009 09:32:28 +0000</pubDate>
		<dc:creator>Ananda</dc:creator>
				<category><![CDATA[(all categories)]]></category>
		<category><![CDATA[Geekiness]]></category>
		<category><![CDATA[Useless Knowledge]]></category>
		<category><![CDATA[probability]]></category>
		<category><![CDATA[R]]></category>
		<category><![CDATA[sampling]]></category>
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		<guid isPermaLink="false">http://news.mrdwab.com/?p=535</guid>
		<description><![CDATA[<p>In my last post about sampling (Simple sampling with R) we were doing simple sampling without replacement&#8211;that is, each item could only be selected once. However, there are times when you want to simulate sampling with replacement. For example, if you wanted to simulate sampling the results of rolling a dice 50 times, your outcomes [...]]]></description>
			<content:encoded><![CDATA[<p>In my last post about sampling (<a href="http://news.mrdwab.com/2009-11-29/simple-sampling-with-r/" title="Simple sampling with R">Simple sampling with R</a>) we were doing simple sampling without replacement&#8211;that is, each item could only be selected once. However, there are times when you want to simulate sampling with replacement. For example, if you wanted to simulate sampling the results of rolling a dice 50 times, your outcomes each time could be a 1, 2, 3, 4, 5 or 6, but 50 is more than 6, so you need to let the software &#8220;replace&#8221; the sample before it takes another sample.</p>
<p>This post explains how to do this with R.</p>
<p><span id="more-535"></span></p>
<pre class="r"><span class="r sym">&gt;</span> <span class="r slc"># ========================================================</span>
<span class="r sym">&gt;</span> <span class="r slc"># Can we sample from things that are not numbers, for</span>
<span class="r sym">&gt;</span> <span class="r slc"># example, pretend we are taking M&amp;Ms out of a jar that</span>
<span class="r sym">&gt;</span> <span class="r slc"># has blue, green, and red M&amp;Ms?</span>
<span class="r sym">&gt;</span> <span class="r slc"># ========================================================</span>
<span class="r sym">&gt;</span> candy <span class="r sym">=</span> <span class="r kwc">c</span><span class="r sym">(</span><span class="r str">&quot;blue&quot;</span><span class="r sym">,</span><span class="r str">&quot;green&quot;</span><span class="r sym">,</span><span class="r str">&quot;red&quot;</span><span class="r sym">)</span>
<span class="r sym">&gt;</span> <span class="r kwc">sample</span><span class="r sym">(</span>candy<span class="r sym">,</span> <span class="r num">20</span><span class="r sym">,</span> <span class="r kwc">replace</span><span class="r sym">=</span><span class="r kwb">T</span><span class="r sym">)</span>
 <span class="r sym">[</span><span class="r num">1</span><span class="r sym">]</span> <span class="r str">&quot;red&quot;</span>   <span class="r str">&quot;red&quot;</span>   <span class="r str">&quot;red&quot;</span>   <span class="r str">&quot;red&quot;</span>   <span class="r str">&quot;blue&quot;</span>  <span class="r str">&quot;green&quot;</span> <span class="r str">&quot;red&quot;</span>
 <span class="r sym">[</span><span class="r num">8</span><span class="r sym">]</span> <span class="r str">&quot;blue&quot;</span>  <span class="r str">&quot;blue&quot;</span>  <span class="r str">&quot;blue&quot;</span>  <span class="r str">&quot;blue&quot;</span>  <span class="r str">&quot;green&quot;</span> <span class="r str">&quot;green&quot;</span> <span class="r str">&quot;green&quot;</span>
<span class="r sym">[</span><span class="r num">15</span><span class="r sym">]</span> <span class="r str">&quot;blue&quot;</span>  <span class="r str">&quot;blue&quot;</span>  <span class="r str">&quot;blue&quot;</span>  <span class="r str">&quot;green&quot;</span> <span class="r str">&quot;blue&quot;</span>  <span class="r str">&quot;red&quot;</span>
<span class="r sym">&gt;</span> <span class="r slc"># ========================================================</span>
<span class="r sym">&gt;</span> <span class="r slc"># &quot;replace=T&quot; is required since there are only three items</span>
<span class="r sym">&gt;</span> <span class="r slc"># in our list. It means that we can sample the same item</span>
<span class="r sym">&gt;</span> <span class="r slc"># more than one time.</span>
<span class="r sym">&gt;</span> <span class="r slc"># ========================================================</span>
<span class="r sym">&gt;</span> <span class="r slc"># ========================================================</span>
<span class="r sym">&gt;</span> <span class="r slc"># What about probability?</span>
<span class="r sym">&gt;</span> <span class="r slc"># ========================================================</span>
<span class="r sym">&gt;</span> <span class="r kwc">sample</span><span class="r sym">(</span>candy<span class="r sym">,</span> <span class="r num">20</span><span class="r sym">,</span> <span class="r kwc">replace</span><span class="r sym">=</span><span class="r kwb">T</span><span class="r sym">,</span> prob<span class="r sym">=</span><span class="r kwc">c</span><span class="r sym">(</span><span class="r num">0.7</span><span class="r sym">,</span> <span class="r num">0.2</span><span class="r sym">,</span> <span class="r num">0.1</span><span class="r sym">))</span>
 <span class="r sym">[</span><span class="r num">1</span><span class="r sym">]</span> <span class="r str">&quot;blue&quot;</span>  <span class="r str">&quot;blue&quot;</span>  <span class="r str">&quot;green&quot;</span> <span class="r str">&quot;blue&quot;</span>  <span class="r str">&quot;blue&quot;</span>  <span class="r str">&quot;blue&quot;</span>  <span class="r str">&quot;blue&quot;</span>
 <span class="r sym">[</span><span class="r num">8</span><span class="r sym">]</span> <span class="r str">&quot;green&quot;</span> <span class="r str">&quot;green&quot;</span> <span class="r str">&quot;blue&quot;</span>  <span class="r str">&quot;blue&quot;</span>  <span class="r str">&quot;blue&quot;</span>  <span class="r str">&quot;blue&quot;</span>  <span class="r str">&quot;blue&quot;</span>
<span class="r sym">[</span><span class="r num">15</span><span class="r sym">]</span> <span class="r str">&quot;green&quot;</span> <span class="r str">&quot;blue&quot;</span>  <span class="r str">&quot;green&quot;</span> <span class="r str">&quot;green&quot;</span> <span class="r str">&quot;green&quot;</span> <span class="r str">&quot;blue&quot;</span>
<span class="r sym">&gt;</span> <span class="r slc"># ========================================================</span>
<span class="r sym">&gt;</span> <span class="r slc"># A second time</span>
<span class="r sym">&gt;</span> <span class="r slc"># ========================================================</span>
<span class="r sym">&gt;</span> <span class="r kwc">sample</span><span class="r sym">(</span>candy<span class="r sym">,</span> <span class="r num">20</span><span class="r sym">,</span> <span class="r kwc">replace</span><span class="r sym">=</span><span class="r kwb">T</span><span class="r sym">,</span> prob<span class="r sym">=</span><span class="r kwc">c</span><span class="r sym">(</span><span class="r num">0.7</span><span class="r sym">,</span> <span class="r num">0.2</span><span class="r sym">,</span> <span class="r num">0.1</span><span class="r sym">))</span>
 <span class="r sym">[</span><span class="r num">1</span><span class="r sym">]</span> <span class="r str">&quot;green&quot;</span> <span class="r str">&quot;blue&quot;</span>  <span class="r str">&quot;green&quot;</span> <span class="r str">&quot;blue&quot;</span>  <span class="r str">&quot;blue&quot;</span>  <span class="r str">&quot;blue&quot;</span>  <span class="r str">&quot;blue&quot;</span>
 <span class="r sym">[</span><span class="r num">8</span><span class="r sym">]</span> <span class="r str">&quot;blue&quot;</span>  <span class="r str">&quot;blue&quot;</span>  <span class="r str">&quot;blue&quot;</span>  <span class="r str">&quot;blue&quot;</span>  <span class="r str">&quot;blue&quot;</span>  <span class="r str">&quot;blue&quot;</span>  <span class="r str">&quot;green&quot;</span>
<span class="r sym">[</span><span class="r num">15</span><span class="r sym">]</span> <span class="r str">&quot;blue&quot;</span>  <span class="r str">&quot;blue&quot;</span>  <span class="r str">&quot;red&quot;</span>   <span class="r str">&quot;green&quot;</span> <span class="r str">&quot;green&quot;</span> <span class="r str">&quot;blue&quot;</span>
<span class="r sym">&gt;</span> <span class="r slc"># ========================================================</span>
<span class="r sym">&gt;</span> <span class="r slc"># Notice how, in general, we have more blues and greens</span>
<span class="r sym">&gt;</span> <span class="r slc"># and almost no reds. You can sort these results too, but</span>
<span class="r sym">&gt;</span> <span class="r slc"># then  you don't see the sequence in which the items</span>
<span class="r sym">&gt;</span> <span class="r slc"># were selected, which might also be interesting. Here</span>
<span class="r sym">&gt;</span> <span class="r slc"># are three more samples, sorted so you can see the effect</span>
<span class="r sym">&gt;</span> <span class="r slc"># that setting the probability has on the outcome.</span>
<span class="r sym">&gt;</span> <span class="r slc"># ========================================================</span>
<span class="r sym">&gt;</span> <span class="r kwc">sort</span><span class="r sym">(</span><span class="r kwc">sample</span><span class="r sym">(</span>candy<span class="r sym">,</span> <span class="r num">20</span><span class="r sym">,</span> <span class="r kwc">replace</span><span class="r sym">=</span><span class="r kwb">T</span><span class="r sym">,</span> prob<span class="r sym">=</span><span class="r kwc">c</span><span class="r sym">(</span><span class="r num">0.7</span><span class="r sym">,</span> <span class="r num">0.2</span><span class="r sym">,</span> <span class="r num">0.1</span><span class="r sym">)))</span>
 <span class="r sym">[</span><span class="r num">1</span><span class="r sym">]</span> <span class="r str">&quot;blue&quot;</span>  <span class="r str">&quot;blue&quot;</span>  <span class="r str">&quot;blue&quot;</span>  <span class="r str">&quot;blue&quot;</span>  <span class="r str">&quot;blue&quot;</span>  <span class="r str">&quot;blue&quot;</span>  <span class="r str">&quot;blue&quot;</span>
 <span class="r sym">[</span><span class="r num">8</span><span class="r sym">]</span> <span class="r str">&quot;blue&quot;</span>  <span class="r str">&quot;blue&quot;</span>  <span class="r str">&quot;blue&quot;</span>  <span class="r str">&quot;blue&quot;</span>  <span class="r str">&quot;blue&quot;</span>  <span class="r str">&quot;blue&quot;</span>  <span class="r str">&quot;blue&quot;</span>
<span class="r sym">[</span><span class="r num">15</span><span class="r sym">]</span> <span class="r str">&quot;blue&quot;</span>  <span class="r str">&quot;blue&quot;</span>  <span class="r str">&quot;green&quot;</span> <span class="r str">&quot;green&quot;</span> <span class="r str">&quot;red&quot;</span>   <span class="r str">&quot;red&quot;</span>
<span class="r sym">&gt;</span> <span class="r kwc">sort</span><span class="r sym">(</span><span class="r kwc">sample</span><span class="r sym">(</span>candy<span class="r sym">,</span> <span class="r num">20</span><span class="r sym">,</span> <span class="r kwc">replace</span><span class="r sym">=</span><span class="r kwb">T</span><span class="r sym">,</span> prob<span class="r sym">=</span><span class="r kwc">c</span><span class="r sym">(</span><span class="r num">0.7</span><span class="r sym">,</span> <span class="r num">0.2</span><span class="r sym">,</span> <span class="r num">0.1</span><span class="r sym">)))</span>
 <span class="r sym">[</span><span class="r num">1</span><span class="r sym">]</span> <span class="r str">&quot;blue&quot;</span>  <span class="r str">&quot;blue&quot;</span>  <span class="r str">&quot;blue&quot;</span>  <span class="r str">&quot;blue&quot;</span>  <span class="r str">&quot;blue&quot;</span>  <span class="r str">&quot;blue&quot;</span>  <span class="r str">&quot;blue&quot;</span>
 <span class="r sym">[</span><span class="r num">8</span><span class="r sym">]</span> <span class="r str">&quot;blue&quot;</span>  <span class="r str">&quot;blue&quot;</span>  <span class="r str">&quot;blue&quot;</span>  <span class="r str">&quot;blue&quot;</span>  <span class="r str">&quot;blue&quot;</span>  <span class="r str">&quot;blue&quot;</span>  <span class="r str">&quot;blue&quot;</span>
<span class="r sym">[</span><span class="r num">15</span><span class="r sym">]</span> <span class="r str">&quot;blue&quot;</span>  <span class="r str">&quot;green&quot;</span> <span class="r str">&quot;red&quot;</span>   <span class="r str">&quot;red&quot;</span>   <span class="r str">&quot;red&quot;</span>   <span class="r str">&quot;red&quot;</span>
<span class="r sym">&gt;</span> <span class="r kwc">sort</span><span class="r sym">(</span><span class="r kwc">sample</span><span class="r sym">(</span>candy<span class="r sym">,</span> <span class="r num">20</span><span class="r sym">,</span> <span class="r kwc">replace</span><span class="r sym">=</span><span class="r kwb">T</span><span class="r sym">,</span> prob<span class="r sym">=</span><span class="r kwc">c</span><span class="r sym">(</span><span class="r num">0.7</span><span class="r sym">,</span> <span class="r num">0.2</span><span class="r sym">,</span> <span class="r num">0.1</span><span class="r sym">)))</span>
 <span class="r sym">[</span><span class="r num">1</span><span class="r sym">]</span> <span class="r str">&quot;blue&quot;</span>  <span class="r str">&quot;blue&quot;</span>  <span class="r str">&quot;blue&quot;</span>  <span class="r str">&quot;blue&quot;</span>  <span class="r str">&quot;blue&quot;</span>  <span class="r str">&quot;blue&quot;</span>  <span class="r str">&quot;blue&quot;</span>
 <span class="r sym">[</span><span class="r num">8</span><span class="r sym">]</span> <span class="r str">&quot;blue&quot;</span>  <span class="r str">&quot;blue&quot;</span>  <span class="r str">&quot;blue&quot;</span>  <span class="r str">&quot;blue&quot;</span>  <span class="r str">&quot;blue&quot;</span>  <span class="r str">&quot;blue&quot;</span>  <span class="r str">&quot;green&quot;</span>
<span class="r sym">[</span><span class="r num">15</span><span class="r sym">]</span> <span class="r str">&quot;green&quot;</span> <span class="r str">&quot;green&quot;</span> <span class="r str">&quot;green&quot;</span> <span class="r str">&quot;green&quot;</span> <span class="r str">&quot;green&quot;</span> <span class="r str">&quot;red&quot;</span>
<span class="r sym">&gt;</span> <span class="r slc"># ========================================================</span>
<span class="r sym">&gt;</span> <span class="r slc"># You can also assign a name to the probability set before</span>
<span class="r sym">&gt;</span> <span class="r slc"># you start your sample instead of typing out the</span>
<span class="r sym">&gt;</span> <span class="r slc"># probabilities each time you take a sample.</span>
<span class="r sym">&gt;</span> <span class="r slc"># ========================================================</span>
<span class="r sym">&gt;</span> ProbCandy <span class="r sym">=</span> <span class="r kwc">c</span><span class="r sym">(</span><span class="r num">0.7</span><span class="r sym">,</span> <span class="r num">0.2</span><span class="r sym">,</span> <span class="r num">0.1</span><span class="r sym">)</span>
<span class="r sym">&gt;</span> <span class="r kwc">sort</span><span class="r sym">(</span><span class="r kwc">sample</span><span class="r sym">(</span>candy<span class="r sym">,</span> <span class="r num">20</span><span class="r sym">,</span> <span class="r kwc">replace</span><span class="r sym">=</span><span class="r kwb">T</span><span class="r sym">,</span> prob<span class="r sym">=</span>ProbCandy<span class="r sym">))</span>
 <span class="r sym">[</span><span class="r num">1</span><span class="r sym">]</span> <span class="r str">&quot;blue&quot;</span>  <span class="r str">&quot;blue&quot;</span>  <span class="r str">&quot;blue&quot;</span>  <span class="r str">&quot;blue&quot;</span>  <span class="r str">&quot;blue&quot;</span>  <span class="r str">&quot;blue&quot;</span>  <span class="r str">&quot;blue&quot;</span>
 <span class="r sym">[</span><span class="r num">8</span><span class="r sym">]</span> <span class="r str">&quot;blue&quot;</span>  <span class="r str">&quot;blue&quot;</span>  <span class="r str">&quot;blue&quot;</span>  <span class="r str">&quot;blue&quot;</span>  <span class="r str">&quot;blue&quot;</span>  <span class="r str">&quot;blue&quot;</span>  <span class="r str">&quot;blue&quot;</span>
<span class="r sym">[</span><span class="r num">15</span><span class="r sym">]</span> <span class="r str">&quot;blue&quot;</span>  <span class="r str">&quot;blue&quot;</span>  <span class="r str">&quot;blue&quot;</span>  <span class="r str">&quot;blue&quot;</span>  <span class="r str">&quot;blue&quot;</span>  <span class="r str">&quot;green&quot;</span>
</pre>
]]></content:encoded>
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		</item>
		<item>
		<title>Simple sampling with R</title>
		<link>http://news.mrdwab.com/2009-11-29/simple-sampling-with-r/</link>
		<comments>http://news.mrdwab.com/2009-11-29/simple-sampling-with-r/#comments</comments>
		<pubDate>Sun, 29 Nov 2009 09:10:08 +0000</pubDate>
		<dc:creator>Ananda</dc:creator>
				<category><![CDATA[Geekiness]]></category>
		<category><![CDATA[Useless Knowledge]]></category>
		<category><![CDATA[code]]></category>
		<category><![CDATA[R]]></category>
		<category><![CDATA[sampling]]></category>
		<category><![CDATA[statistics]]></category>

		<guid isPermaLink="false">http://news.mrdwab.com/?p=530</guid>
		<description><![CDATA[<p>I mentioned in an earlier post (Am I inconsistent?) that I got interested in R because Amy had asked me to help her with some sampling at one point. Since that was my starting point, I thought I would share some of my experiments with you.</p>
<p>In this post:</p>

Simple random sampling
Simple random sampling with a seed
Sorting [...]]]></description>
			<content:encoded><![CDATA[<p>I mentioned in an earlier post (<a href="http://news.mrdwab.com/2009-11-15/am-i-inconsistent/" title="Am I inconsistent?">Am I inconsistent?</a>) that I got interested in R because Amy had asked me to help her with some sampling at one point. Since that was my starting point, I thought I would share some of my experiments with you.</p>
<p>In this post:</p>
<ol>
<li>Simple random sampling</li>
<li>Simple random sampling with a seed</li>
<li>Sorting your sample</li>
</ol>
<p><span id="more-530"></span></p>
<pre class="r"><span class="r sym">&gt;</span> <span class="r slc"># ========================================================</span>
<span class="r sym">&gt;</span> <span class="r slc"># Sample 30 from 1 to 300</span>
<span class="r sym">&gt;</span> <span class="r slc"># ========================================================</span>
<span class="r sym">&gt;</span> <span class="r kwc">sample</span><span class="r sym">(</span><span class="r num">300</span><span class="r sym">,</span> <span class="r num">30</span><span class="r sym">)</span>
 <span class="r sym">[</span><span class="r num">1</span><span class="r sym">]</span>  <span class="r num">44 200 258 290 165 132 287  70 118 241  11 191 194 192</span>
<span class="r sym">[</span><span class="r num">15</span><span class="r sym">]</span> <span class="r num">127 143  55 217  49 130 263 203  26  46 251 142  72 107</span>
<span class="r sym">[</span><span class="r num">29</span><span class="r sym">]</span>  <span class="r num">77  31</span>
<span class="r sym">&gt;</span> <span class="r slc"># ========================================================</span>
<span class="r sym">&gt;</span> <span class="r slc"># A second time</span>
<span class="r sym">&gt;</span> <span class="r slc"># ========================================================</span>
<span class="r sym">&gt;</span> <span class="r kwc">sample</span><span class="r sym">(</span><span class="r num">300</span><span class="r sym">,</span> <span class="r num">30</span><span class="r sym">)</span>
 <span class="r sym">[</span><span class="r num">1</span><span class="r sym">]</span> <span class="r num">101 108  25 289 257 102 110  45 173 197 202 168 134 225</span>
<span class="r sym">[</span><span class="r num">15</span><span class="r sym">]</span> <span class="r num">184 155 226  68  86  51  90 204 159 263 245 201 213 169</span>
<span class="r sym">[</span><span class="r num">29</span><span class="r sym">]</span> <span class="r num">127  11</span>
<span class="r sym">&gt;</span> <span class="r slc"># ========================================================</span>
<span class="r sym">&gt;</span> <span class="r slc"># What about a replicable sample?</span>
<span class="r sym">&gt;</span> <span class="r slc"># ========================================================</span>
<span class="r sym">&gt;</span> <span class="r kwc">set.seed</span><span class="r sym">(</span><span class="r num">123</span><span class="r sym">)</span>
<span class="r sym">&gt;</span> <span class="r kwc">sample</span><span class="r sym">(</span><span class="r num">300</span><span class="r sym">,</span> <span class="r num">30</span><span class="r sym">)</span>
 <span class="r sym">[</span><span class="r num">1</span><span class="r sym">]</span>  <span class="r num">87 236 122 263 279  14 156 262 162 133 278 132 196 165</span>
<span class="r sym">[</span><span class="r num">15</span><span class="r sym">]</span>  <span class="r num">30 257  70  12  93 269 250 194 179 276 181 195 150 163</span>
<span class="r sym">[</span><span class="r num">29</span><span class="r sym">]</span>  <span class="r num">79  40</span>
<span class="r sym">&gt;</span> <span class="r slc"># ========================================================</span>
<span class="r sym">&gt;</span> <span class="r slc"># A second time</span>
<span class="r sym">&gt;</span> <span class="r slc"># ========================================================</span>
<span class="r sym">&gt;</span> <span class="r kwc">set.seed</span><span class="r sym">(</span><span class="r num">123</span><span class="r sym">)</span>
<span class="r sym">&gt;</span> <span class="r kwc">sample</span><span class="r sym">(</span><span class="r num">300</span><span class="r sym">,</span> <span class="r num">30</span><span class="r sym">)</span>
 <span class="r sym">[</span><span class="r num">1</span><span class="r sym">]</span>  <span class="r num">87 236 122 263 279  14 156 262 162 133 278 132 196 165</span>
<span class="r sym">[</span><span class="r num">15</span><span class="r sym">]</span>  <span class="r num">30 257  70  12  93 269 250 194 179 276 181 195 150 163</span>
<span class="r sym">[</span><span class="r num">29</span><span class="r sym">]</span>  <span class="r num">79  40</span>
<span class="r sym">&gt;</span> <span class="r slc"># ========================================================</span>
<span class="r sym">&gt;</span> <span class="r slc"># Can we sort the output to make the result easy to see?</span>
<span class="r sym">&gt;</span> <span class="r slc"># ========================================================</span>
<span class="r sym">&gt;</span> <span class="r kwc">set.seed</span><span class="r sym">(</span><span class="r num">123</span><span class="r sym">)</span>
<span class="r sym">&gt;</span> <span class="r kwc">sort</span><span class="r sym">(</span><span class="r kwc">sample</span><span class="r sym">(</span><span class="r num">300</span><span class="r sym">,</span> <span class="r num">30</span><span class="r sym">))</span>
 <span class="r sym">[</span><span class="r num">1</span><span class="r sym">]</span>  <span class="r num">12  14  30  40  70  79  87  93 122 132 133 150 156 162</span>
<span class="r sym">[</span><span class="r num">15</span><span class="r sym">]</span> <span class="r num">163 165 179 181 194 195 196 236 250 257 262 263 269 276</span>
<span class="r sym">[</span><span class="r num">29</span><span class="r sym">]</span> <span class="r num">278 279</span>
</pre>
<h2>What is &#8220;set.seed&#8221; for?</h2>
<p>The &#8220;set.seed&#8221; function in R can help when you want to verify or replicate a sample. In this case, the seed I used was &#8220;123&#8243; (but it can be any number you want it to be). So, if you run the same sample with the same seed in R, you should get the same results as me. If you ran the same sample without a seed, each time, you would get a different result.</p>
<p>Disclaimer: I&#8217;m not by any measures an expert in R. I have just been experimenting with the software. By the way, if you want to experiment with any of the code I mention here, you can first <a href="http://www.r-project.org">download R</a>, copy the code, and in R, go to &#8220;Edit > Paste commands only&#8221;.</p>
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