document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=515085>Question 515085</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Asking again because email was entered wrong\r<BR>\n" );
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document.write( "Find the probability of the given value of x<BR>\n" );
document.write( "mean = 28 standard deviation= 6.5 and x is equal to or less than 20<BR>\n" );
document.write( "We are using the standard distribution z chart and I know it is plus or minus .5<BR>\n" );
document.write( "However, I do not get the right answers<BR>\n" );
document.write( "We are using z=x-u/o formula </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #343710 by </B><A HREF=/tutors/aboutme.mpl?userid=drcole><B>drcole(72)</B></A><img src=\"/images/stars/stars-08.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=drcole><IMG SRC=http://www.algebra.com/images/aboutme-small.gif BORDER=0 alt=\"About Me\" VALIGN=MIDDLE></A>&nbsp;<IMG SRC=http://www.algebra.com/images/nopicture.jpg><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=343710\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> You have a random variable x that is normally distributed, with mean <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=mu+=+28 ALIGN=MIDDLE  ALT=\"mu+=+28\"   DISABLED_style= \"max-width:90%; height:auto;\" > and standard deviation <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=sigma+=+6.5 ALIGN=MIDDLE  ALT=\"sigma+=+6.5\"   DISABLED_style= \"max-width:90%; height:auto;\" > (these lowercase Greek letters are pronounced \"mu\" and \"sigma\" respectively.  You need to figure out the probability that x is less than or equal to 20.  You have the standard normal distribution table.  How do we put this all together?\r<BR>\n" );
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document.write( "The idea here is we can change this problem on a non-standard normal distribution to one on the standard normal distribution by computing the z-score of 20.  The z-score of 20 is simply the number of standard deviations 20 is above or below the mean.  The formula you have tells you the z-score:\r<BR>\n" );
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document.write( "z-score = <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28x+-+mu%29%2Fsigma ALIGN=MIDDLE  ALT=\"%28x+-+mu%29%2Fsigma\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "Let's apply this formula:\r<BR>\n" );
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document.write( "z-score of 20 = <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%2820+-+28%29%2F6.5+=+-1.23 ALIGN=MIDDLE  ALT=\"%2820+-+28%29%2F6.5+=+-1.23\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "This makes sense: 20 is more than one, but less than two standard deviations below the mean of 28, and a z-score of -1.23 is consistent with this observation.\r<BR>\n" );
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document.write( "It turns out that, if x is normally distributed, then the probability that x <= 20 is exactly the same as the probability that z, a random variable on the standard normal distribution, is less than or equal to the z-score of 20, or -1.23:\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=P%28x+%3C=+20%29+=+P%28z+%3C=+-1.23%29 ALIGN=MIDDLE  ALT=\"P%28x+%3C=+20%29+=+P%28z+%3C=+-1.23%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "You can now use your table to find the probability that z <= -1.23.  If your table gives probabilities for positive and negative z-scores, just look up -1.23 and read off the number.  You should get:\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=P%28z+%3C=+-1.23%29+=0.1093 ALIGN=MIDDLE  ALT=\"P%28z+%3C=+-1.23%29+=0.1093\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "or something very close to it (depending on how the values were rounded).  If your table only gives probabilities for positive z-scores, then you can still get the right answer by observing that the standard normal distribution is symmetric: its bell curve shape is exactly the same on both sides of the mean.  That means that the probability that z <= -1.23 is exactly the same as the probability that z >= 1.23, its mirror image.  Since the probabilities that z <= 1.23 and z >= 1.23 should add to 1, we get that:\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=P%28z+%3C=+1.23%29+%2B+P%28z+%3E=+1.23%29+=+1 ALIGN=MIDDLE  ALT=\"P%28z+%3C=+1.23%29+%2B+P%28z+%3E=+1.23%29+=+1\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=P%28z+%3E=+1.23%29+=+1+-+P%28z+%3C=+1.23%29 ALIGN=MIDDLE  ALT=\"P%28z+%3E=+1.23%29+=+1+-+P%28z+%3C=+1.23%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > (solving for P(z >= 1.23))<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=P%28z+%3C=+-1.23%29+=+1+-+P%28z+%3C=+1.23%29 ALIGN=MIDDLE  ALT=\"P%28z+%3C=+-1.23%29+=+1+-+P%28z+%3C=+1.23%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > (remembering that <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=P%28z+%3C=+-1.23%29+=+P%28z+%3E=+1.23%29 ALIGN=MIDDLE  ALT=\"P%28z+%3C=+-1.23%29+=+P%28z+%3E=+1.23%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >)\r<BR>\n" );
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document.write( "So we look up the probability that z <= 1.23 in our table:\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=P%28z+%3C=+1.23%29+=+0.8907 ALIGN=MIDDLE  ALT=\"P%28z+%3C=+1.23%29+=+0.8907\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "and then subtract the value we got from the table from 1 to get our answer:\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=P%28z+%3C=+-1.23%29+=+1+-+0.8907+=+0.1093 ALIGN=MIDDLE  ALT=\"P%28z+%3C=+-1.23%29+=+1+-+0.8907+=+0.1093\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "which is exactly what we got before.  Since the probability that x <= 20 is equal to the probability that z <= -1.23, this is our answer.\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=P%28x+%3C=+20%29+=+0.1093 ALIGN=MIDDLE  ALT=\"P%28x+%3C=+20%29+=+0.1093\"   DISABLED_style= \"max-width:90%; height:auto;\" > </SPAN>\n" );
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