document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=919955>Question 919955</A>: <I><SPAN STYLE=\"word-break: break-all;\"> THE AVERAGE AGE OF C.E.Os, IS 56 YEARS. ASSUME THE VARIABLE IS NORMALLY DISTRIBUTED. IF THE STANDARD DEVIATION IS 4 YEARS, FIND THE YOUNGEST AND THE OLDEST PERSON WHO QUALIFIES TO BE A C.E.O.\r<BR>\n" );
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document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #558074 by </B><A HREF=/tutors/aboutme.mpl?userid=Theo><B>Theo(13342)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=Theo><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/cgi-bin/show-face.mpl?user=Theo><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=558074\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> you cannot absolutely find the youngest person or the oldest person who qualifies to be a ceo using the normal distribution.\r<BR>\n" );
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document.write( "you can only find the probability that the youngest person will be less than a certain age and the probability that the oldest person will be greater than a certain age.\r<BR>\n" );
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document.write( "what you need to be asking is something like:\r<BR>\n" );
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document.write( "what is the youngest possible age and oldest possible age so that the probability that a person can be ceo of a corporation is 99% or some other figure, assuming the average age is 56 years old and the standard deviation is 4 years.\r<BR>\n" );
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document.write( "now you have something you can look up in a table or use a calculator to find.\r<BR>\n" );
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document.write( "for example:\r<BR>\n" );
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document.write( "at 99% confidence interval, the z-score will be between -2.576 and 2.576 rounded up to 3 decimal places.\r<BR>\n" );
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document.write( "the probability of a z-score being between -2.576 and 2.576 is at least 99%.\r<BR>\n" );
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document.write( "now you want to translate the z-score to a raw score.\r<BR>\n" );
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document.write( "that is done as follows:\r<BR>\n" );
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document.write( "the formula is:\r<BR>\n" );
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document.write( "z = (x-m)/s\r<BR>\n" );
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document.write( "z is the z-score.<BR>\n" );
document.write( "x is the raw score<BR>\n" );
document.write( "m is the mean<BR>\n" );
document.write( "s is the standard deviation.\r<BR>\n" );
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document.write( "in your problem:\r<BR>\n" );
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document.write( "z = plus or minus 2.576.<BR>\n" );
document.write( "x = what you want to find.<BR>\n" );
document.write( "m = the average age of a ceo which is 56 years of age.<BR>\n" );
document.write( "s is the standard deviation which is 4 years.\r<BR>\n" );
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document.write( "you want to solve for x.\r<BR>\n" );
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document.write( "start with:\r<BR>\n" );
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document.write( "z = (x-m)/s\r<BR>\n" );
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document.write( "multiply both sides of this equation by s to get:\r<BR>\n" );
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document.write( "z*s = x-m\r<BR>\n" );
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document.write( "add m to both sides of this equation to get:\r<BR>\n" );
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document.write( "z*s + m = x which is the same as:\r<BR>\n" );
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document.write( "x = z*s + m\r<BR>\n" );
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document.write( "replace z with 2.576 and s with 4 and m with 56 and you get:\r<BR>\n" );
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document.write( "2.576 * 4 + 56 = 66.304 which you can round up to 67.\r<BR>\n" );
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document.write( "replace z with -2.576 and s with 4 and m with 56 and you get:\r<BR>\n" );
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document.write( "-2.576 * 4 + 56 = 45.696 which you can round down to 45.\r<BR>\n" );
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document.write( "rounding the low score down and the high score up is a conservative estimate that ensures you will be at least 99% confident the age will be between those limits.\r<BR>\n" );
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document.write( "there is at least a 99% probability that the age of a ceo will be between 45 and 67 years of age.\r<BR>\n" );
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document.write( "that still leaves at most 1% of ceos that could either be above 67 or below 45 years of age.\r<BR>\n" );
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document.write( "if you want to be more certain, then raise the confidence interval.\r<BR>\n" );
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document.write( "assume you want to be 99.999% sure that the age of the ceo will be within certain limits.\r<BR>\n" );
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document.write( "that generates a different z-score.\r<BR>\n" );
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document.write( "99.999% probability generates a z-score of plus or minus 4.418 rounded up to 3 decimal places.\r<BR>\n" );
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document.write( "the same formula applies.\r<BR>\n" );
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document.write( "x = z*s + m\r<BR>\n" );
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document.write( "on the high side, you will get x = 4.418 * 4 + 56 = 73.672 which you can round up to 74.\r<BR>\n" );
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document.write( "on the low side, you will get x = -4.418 * 4 + 56 = 38.328 which you can round down to 38.\r<BR>\n" );
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document.write( "there is at least 99.999% probability that a ceo will be between the ages of 38 and 74.\r<BR>\n" );
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document.write( "that doesn't mean there can't be a ceo younger than 38 years of age and doesn't mean that there can't be a ceo older than 74 years of age.\r<BR>\n" );
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document.write( "it just means that the probability of that happening is very low.\r<BR>\n" );
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document.write( "in fact, the probability that a ceo will be less than 38 years of age or more than 74 years of age is at most .001%\r<BR>\n" );
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document.write( "if you want to see this visually, the following pictures might help.\r<BR>\n" );
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document.write( "99% probability:\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" src = \"http://theo.x10hosting.com/2014/110404.jpg\" alt=\"$$$\" </>\r<BR>\n" );
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document.write( "99.999% probability:\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" src = \"http://theo.x10hosting.com/2014/110405.jpg\" alt=\"$$$\" </>\r<BR>\n" );
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document.write( "the numbers are slightly different due to rounding errors and the fact that i took a conservative view of the z-score required while the calculator simply rounded up or down depending on whether the next digit was 5 or more for up and less than 5 for down.\r<BR>\n" );
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document.write( "round the low numbers shown in the picture down to the nearest integer and the high number shown in the picture up to the nearest integer and we agree.\r<BR>\n" );
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document.write( "for 99%, i got 45 to 67 and the picture shows 45 to 67.<BR>\n" );
document.write( "for 99.999%, i got 38 to 74 and the picture shows 38 to 74.\r<BR>\n" );
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document.write( "any differences would have been due to rounding errors and assumptions.\r<BR>\n" );
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