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  <name xmlns:md="http://cnx.rice.edu/mdml/0.4" xmlns:m="http://www.w3.org/1998/Math/MathML" xmlns:bib="http://bibtexml.sf.net/">Normal Distribution: Areas to the Left and Right of x</name>
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      <md:surname xmlns:m="http://www.w3.org/1998/Math/MathML" xmlns:bib="http://bibtexml.sf.net/">Connexions</md:surname>
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    <md:keyword xmlns:m="http://www.w3.org/1998/Math/MathML" xmlns:bib="http://bibtexml.sf.net/">elementary</md:keyword>
    <md:keyword xmlns:m="http://www.w3.org/1998/Math/MathML" xmlns:bib="http://bibtexml.sf.net/">statistics</md:keyword>
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  <md:abstract xmlns:m="http://www.w3.org/1998/Math/MathML" xmlns:bib="http://bibtexml.sf.net/"/>
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  <content xmlns:md="http://cnx.rice.edu/mdml/0.4" xmlns:m="http://www.w3.org/1998/Math/MathML" xmlns:bib="http://bibtexml.sf.net/">
    <para xmlns:md="http://cnx.rice.edu/mdml/0.4" xmlns:m="http://www.w3.org/1998/Math/MathML" xmlns:bib="http://bibtexml.sf.net/" id="delete_me">The arrow in the graph below points to the area to the left of <m:math><m:mi>x</m:mi></m:math>. This area is
represented by the probability 
<emphasis xmlns:md="http://cnx.rice.edu/mdml/0.4" xmlns:m="http://www.w3.org/1998/Math/MathML" xmlns:bib="http://bibtexml.sf.net/">
<m:math>
<m:reln><m:lt/>
<m:mrow>
<m:mi>P</m:mi>
<m:mo>(</m:mo>
<m:mi>X</m:mi>
</m:mrow>
<m:mrow>
<m:mi>x</m:mi>
<m:mo>)</m:mo>
</m:mrow>
</m:reln>
</m:math>. 
</emphasis>
Normal tables, computers, and calculators
provide or calculate the probability 
<m:math>
<m:reln><m:lt/>
<m:mrow>
<m:mi>P</m:mi>
<m:mo>(</m:mo>
<m:mi>X</m:mi>
</m:mrow>
<m:mrow>
<m:mi>x</m:mi>
<m:mo>)</m:mo>
</m:mrow>
</m:reln>
</m:math>.</para><para xmlns:md="http://cnx.rice.edu/mdml/0.4" xmlns:m="http://www.w3.org/1998/Math/MathML" xmlns:bib="http://bibtexml.sf.net/" id="element-741"><media xmlns:md="http://cnx.rice.edu/mdml/0.4" xmlns:m="http://www.w3.org/1998/Math/MathML" xmlns:bib="http://bibtexml.sf.net/" type="image/png" src="normdist_arealrx1.png">
<param name="alt" value="Normal distribution curve with a x value on the x-axis. The x-axis is equal to X. A vertical upward line extends from point x to the curve and the probability area occurs from the beginning of the curve to point x."/>

<param name="print-width" value="3in"/>
</media></para><para xmlns:md="http://cnx.rice.edu/mdml/0.4" xmlns:m="http://www.w3.org/1998/Math/MathML" xmlns:bib="http://bibtexml.sf.net/" id="element-97"><emphasis xmlns:md="http://cnx.rice.edu/mdml/0.4" xmlns:m="http://www.w3.org/1998/Math/MathML" xmlns:bib="http://bibtexml.sf.net/">
The area to the right is then
<m:math>
<m:reln><m:gt/>
<m:mrow>
<m:mi>P</m:mi>
<m:mo>(</m:mo>
<m:mi>X</m:mi>
</m:mrow>
<m:mrow>
<m:mi>x</m:mi>
<m:mo>)</m:mo>
</m:mrow>
</m:reln>
<m:mo>=</m:mo>
<m:reln><m:lt/>
<m:mrow>
<m:mn>1</m:mn>
<m:mo>-</m:mo>
<m:mi>P</m:mi>
<m:mo>(</m:mo>
<m:mi>X</m:mi>
</m:mrow>
<m:mrow>
<m:mi>x</m:mi>
<m:mo>)</m:mo>
</m:mrow>
</m:reln>
</m:math>.
</emphasis></para><para xmlns:md="http://cnx.rice.edu/mdml/0.4" xmlns:m="http://www.w3.org/1998/Math/MathML" xmlns:bib="http://bibtexml.sf.net/" id="element-521">Remember, 
<m:math>
<m:reln><m:lt/>
<m:mrow>
<m:mi>P</m:mi>
<m:mo>(</m:mo>
<m:mi>X</m:mi>
</m:mrow>
<m:mrow>
<m:mi>x</m:mi>
<m:mo>)</m:mo>
</m:mrow>
</m:reln>
<m:mo>=</m:mo>
</m:math>

<emphasis xmlns:md="http://cnx.rice.edu/mdml/0.4" xmlns:m="http://www.w3.org/1998/Math/MathML" xmlns:bib="http://bibtexml.sf.net/">Area to the left</emphasis> of the vertical line through <m:math><m:mi>x</m:mi></m:math>.</para><para xmlns:md="http://cnx.rice.edu/mdml/0.4" xmlns:m="http://www.w3.org/1998/Math/MathML" xmlns:bib="http://bibtexml.sf.net/" id="element-732"><m:math>
<m:reln><m:gt/>
<m:mrow>
<m:mi>P</m:mi>
<m:mo>(</m:mo>
<m:mi>X</m:mi>
</m:mrow>
<m:mrow>
<m:mi>x</m:mi>
<m:mo>)</m:mo>
</m:mrow>
</m:reln>
<m:mo>=</m:mo>
<m:reln><m:lt/>
<m:mrow>
<m:mn>1</m:mn>
<m:mo>-</m:mo>
<m:mi>P</m:mi>
<m:mo>(</m:mo>
<m:mi>X</m:mi>
</m:mrow>
<m:mrow>
<m:mi>x</m:mi>
<m:mo>)</m:mo>
</m:mrow>
</m:reln>
<m:mo>=</m:mo>
</m:math>.
<emphasis xmlns:md="http://cnx.rice.edu/mdml/0.4" xmlns:m="http://www.w3.org/1998/Math/MathML" xmlns:bib="http://bibtexml.sf.net/">Area to the right</emphasis>
of the vertical line through <m:math><m:mi>x</m:mi></m:math></para><para xmlns:md="http://cnx.rice.edu/mdml/0.4" xmlns:m="http://www.w3.org/1998/Math/MathML" xmlns:bib="http://bibtexml.sf.net/" id="element-974"><m:math>
<m:reln><m:lt/>
<m:mrow>
<m:mi>P</m:mi>
<m:mo>(</m:mo>
<m:mi>X</m:mi>
</m:mrow>
<m:mrow>
<m:mi>x</m:mi>
<m:mo>)</m:mo>
</m:mrow>
</m:reln>
</m:math>
is the same as
 <m:math>
<m:reln><m:leq/>
<m:mrow>
<m:mi>P</m:mi>
<m:mo>(</m:mo>
<m:mi>X</m:mi>
</m:mrow>
<m:mrow>
<m:mi>x</m:mi>
<m:mo>)</m:mo>
</m:mrow>
</m:reln>
</m:math>
and
 <m:math>
<m:reln><m:gt/>
<m:mrow>
<m:mi>P</m:mi>
<m:mo>(</m:mo>
<m:mi>X</m:mi>
</m:mrow>
<m:mrow>
<m:mi>x</m:mi>
<m:mo>)</m:mo>
</m:mrow>
</m:reln>
</m:math>
is the same as
 <m:math>
<m:reln><m:geq/>
<m:mrow>
<m:mi>P</m:mi>
<m:mo>(</m:mo>
<m:mi>X</m:mi>
</m:mrow>
<m:mrow>
<m:mi>x</m:mi>
<m:mo>)</m:mo>
</m:mrow>
</m:reln>
</m:math>
for continuous distributions.</para>   
  </content>
  
</document>
