document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1206525>Question 1206525</A>: <I><SPAN STYLE=\"word-break: break-all;\"> A company produces steel rods. The lengths of the steel rods are normally distributed with a mean of 257.2-cm and a standard deviation of 1.5-cm. For shipment, 12 steel rods are bundled together.\r<BR>\n" );
document.write( "\n" );
document.write( "Find the probability that the average length of a randomly selected bundle of steel rods is between 257.7-cm and 258.1-cm.<BR>\n" );
document.write( "P(257.7-cm < M < 258.1-cm) = \r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "Enter your answer as a number accurate to 4 decimal places. Answers obtained using exact z-scores or z-scores rounded to 3 decimal places are accepted.<BR>\n" );
document.write( " </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #844086 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=844086\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> mean is 257.2 cm.<BR>\n" );
document.write( "standard deviation is 1.5 cm<BR>\n" );
document.write( "sample size is 12.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "standard error = population standard deviation / sqrt(sample size) = 1.5 / sqrt(12) = .433013.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "you want to know the probability that the mean of the sample will be between 257.7 and 258.1 cm.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "it appears the problem is looking for z-score results, so we'll go with that.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "z-score formula is z = (x - m) / s\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "z is the z-score<BR>\n" );
document.write( "x is the sample mean<BR>\n" );
document.write( "m is the population mean<BR>\n" );
document.write( "s is the standard error\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "standard error = standard deviation / sqrt(sample size) = 1.5 / sqrt(12) = .433013 rounded to 6 decimal places.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "you want to find the z-scores for sample means of 257.7 and 258.1.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "z-score formula for sample mean of 257.7 is z = (257.7 - 257.2) / .433013 = 1.1547.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "z-score formula for sample mean of 258.1 is z = (258.1 - 257.2) / .433013 = 2.0785.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "you can use a z-score calculator, such as the one at <a  rel=nofollow HREF= \"https://davidmlane.com/hyperstat/z_table.html\" target = \"_blank\">https://davidmlane.com/hyperstat/z_table.html</a>\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "use of that calculator tells you that the area between those 2 z-score is equal to .1053.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "that's the probability that a sample of size 12 will have a mean length of between 257.7 and 258.1 centimeters.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "here's what that looks like.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" src = \"http://theo.x10hosting.com/2024/031803.jpg\">\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( " </SPAN>\n" );
document.write( "</TD></TR></TABLE>\n" );