document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=358105>Question 358105</A>: <I><SPAN STYLE=\"word-break: break-all;\"> If the mean and standard deviation of serum iron values for healthy men are 116 and 14 micrograms per 100 mL, respectively, find the probability that a random sample of 58 healthy men will yield a mean of at most 120 micrograms per 100 mL.\r<BR>\n" );
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document.write( "I know that N = 58...I think...and I don't know if this makes sense but is 14/sqrt(58) on the right track?<BR>\n" );
document.write( "If you could please help, that would be extremely appreciated, I am just completely lost! THANK YOU SO MUCH! </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #255628 by </B><A HREF=/tutors/aboutme.mpl?userid=jrfrunner><B>jrfrunner(365)</B></A><img src=\"/images/stars/stars-10.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=jrfrunner><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=jrfrunner><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=255628\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> you are given the following<BR>\n" );
document.write( "The population mean <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=mu=116 ALIGN=MIDDLE  ALT=\"mu=116\"   DISABLED_style= \"max-width:90%; height:auto;\" > and population standard deviation <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=sigma=14 ALIGN=MIDDLE  ALT=\"sigma=14\"   DISABLED_style= \"max-width:90%; height:auto;\" >, the population distribution is not known<BR>\n" );
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document.write( "also you sample n=58 healthy men <BR>\n" );
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document.write( "You want to know \r<BR>\n" );
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document.write( "P(xbar<=120)<BR>\n" );
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document.write( "you are left with a dilemma.  You have partial information, <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=mu ALIGN=MIDDLE  ALT=\"mu\"   DISABLED_style= \"max-width:90%; height:auto;\" > and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=sigma ALIGN=MIDDLE  ALT=\"sigma\"   DISABLED_style= \"max-width:90%; height:auto;\" > but no knowledge of the parent population.  What do you do???<BR>\n" );
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document.write( "Well you rely on the Central Limit Theorem that says:<BR>\n" );
document.write( "If you are analzying sample averages \"and\"  the sample sizes used for those averages is large (ie over 30), the distribution of the sample averages will resemble a normal curve, specially as the sample sizes get larger.<BR>\n" );
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document.write( "Since n=56 can be consider large and you are analzying sample averages you get a free pass to use the normal distribution for your problem<BR>\n" );
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document.write( "Your next step would be to standardize the sample averages you are interested in,  xbar<=120.   Standardize implies you change the values so that they mimick coming from a distribution having <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=mu ALIGN=MIDDLE  ALT=\"mu\"   DISABLED_style= \"max-width:90%; height:auto;\" >=0 instead of <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=mu ALIGN=MIDDLE  ALT=\"mu\"   DISABLED_style= \"max-width:90%; height:auto;\" >=116 and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=sigma ALIGN=MIDDLE  ALT=\"sigma\"   DISABLED_style= \"max-width:90%; height:auto;\" >=1 instead of 14.  How do you do that?<BR>\n" );
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document.write( "To standarize you use the transformation <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=Z=%28xbar+-mu%29%2Fstderror ALIGN=MIDDLE  ALT=\"Z=%28xbar+-mu%29%2Fstderror\"   DISABLED_style= \"max-width:90%; height:auto;\" >, where the standard error = <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=sigma%2Fsqrt%28n%29 ALIGN=MIDDLE  ALT=\"sigma%2Fsqrt%28n%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >.  you have to use standard error because you are analysing sample averages xbar, not individual valus x.<BR>\n" );
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document.write( "So your problem P(Xbar<=120) becomes  <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=P%28Z%3C=%28120-116%29%2F%2814%2Fsqrt%2858%29%29%29 ALIGN=MIDDLE  ALT=\"P%28Z%3C=%28120-116%29%2F%2814%2Fsqrt%2858%29%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > or P(Z<=2.176)<BR>\n" );
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document.write( "from here you can either use book tables, statistical calculators or excel among many other tools to find this probability.\r<BR>\n" );
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document.write( "answer=0.9852 </SPAN>\n" );
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