document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=208486>Question 208486</A>: <I><SPAN STYLE=\"word-break: break-all;\"> In a normal distribution with a mean and a std deviation, how do you find the probability of selecting 3 random units, whose total average exceeds a certain value?\r<BR>\n" );
document.write( "\n" );
document.write( "For example, normal distribution with mean of 127, and SD of 3.1:  what is the probability that if three random units are selected, their total average (add up the values and divide by three), is greater than 130? </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #157708 by </B><A HREF=/tutors/aboutme.mpl?userid=stanbon><B>stanbon(75887)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=stanbon><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=157708\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> In a normal distribution with a mean and a std deviation, how do you find the probability of selecting 3 random units, whose total average exceeds a certain value? <BR>\n" );
document.write( "For example, normal distribution with mean of 127, and SD of 3.1: what is the probability that if three random units are selected, their total average (add up the values and divide by three), is greater than 130?<BR>\n" );
document.write( "-----------------<BR>\n" );
document.write( "That average you are referring to is the sample mean.<BR>\n" );
document.write( "----<BR>\n" );
document.write( "Since the population mean is 127, the mean of the sample means is 127.<BR>\n" );
document.write( "Since the SD of the population is 3.1, the SD of the sample means is 3.1/sqrt(3)<BR>\n" );
document.write( "----<BR>\n" );
document.write( "These facts are known because of the \"Central Limit Theorem\".<BR>\n" );
document.write( "-----------------------<BR>\n" );
document.write( "So find the z-value of 130:<BR>\n" );
document.write( "z(130) = (130-127)/[3.1/sqrt(3)] = 1.676<BR>\n" );
document.write( "---<BR>\n" );
document.write( "This tells you that 130 is 1.676 standard deviations to the right of the<BR>\n" );
document.write( "mean of the sample means, which is 127.<BR>\n" );
document.write( "9<BR>\n" );
document.write( "Then P(a sample mean could be greater than 130) = P(z> 1.676) = 0.0469<BR>\n" );
document.write( "-------------------------------------------<BR>\n" );
document.write( "Cheers,<BR>\n" );
document.write( "Stan H.<BR>\n" );
document.write( "PS. If you have any response to this email stanbon@comcast.net as the <BR>\n" );
document.write( "feedback function of algebra.com is not working.<BR>\n" );
document.write( " </SPAN>\n" );
document.write( "</TD></TR></TABLE>\n" );