document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1121793>Question 1121793</A>: <I><SPAN STYLE=\"word-break: break-all;\"> The accuracy of a cutting machine is such that the mean length is the length it is set to and the standard deviation is 2.2mm. The lengths follow a normal distribution. The engineer wants to make sure that no more than 20% of the lengths cut are longer than 400mm. (It is less of a problem if the lengths are a little short.) What should she set the mean length to be?\r<BR>\n" );
document.write( "\n" );
document.write( "Hint: Find the value of z for which Pr(Z>z)=0.20 then use the z-score formula to calculate the mean.\r<BR>\n" );
document.write( "\n" );
document.write( "Do not give the units in your answer; just give the number.\r<BR>\n" );
document.write( "\n" );
document.write( "Give the answer to two decimal places.\r<BR>\n" );
document.write( "\n" );
document.write( "Thank you so much in advance<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 #737802 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=737802\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> if you want 20% to the right of the critical z-score, you can use the following online calculator to give you the resulting z-score.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "that's the online z-score calculator that i used.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "<a  rel=nofollow HREF= \"http://davidmlane.com/hyperstat/z_table.html\" target = \"_blank\">http://davidmlane.com/hyperstat/z_table.html</a>\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "the z-score table i used can be found at <a  rel=nofollow HREF= \"http://www.z-table.com/\" target = \"_blank\">http://www.z-table.com/</a>\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "use of this calculator will tell you that the critical z-score is .841.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "my straight line interpolation from the use of the z-score table gets me a z-score of .8417857143 which equals .842 rounded to 3 decimal places.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "my TI-84-Plus calculator tells me the z-score is .8416212335 which equals .842 rounded to 3 decimal places.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "what this points out is that you might get a different critical z-score depending on the method or calculator used to get that critical z-score.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "either way, you will get a reasonably accurate result.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "i went with critical z-scores of .841 and .842 and .84 to see what the difference would be.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "the formula used to find the mean is derived from the z-score formula shown below.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "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 raw score<BR>\n" );
document.write( "m is the mean<BR>\n" );
document.write( "s is the standard deviation.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "solve for the mean in this formula as follows:\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "start with z = (x - m) / s<BR>\n" );
document.write( "multiply both sides by s to get z * s =  x - m<BR>\n" );
document.write( "add m to both sides of this formula and subtract z * s from both sides of this formula to get:<BR>\n" );
document.write( "m = x - z * s\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "m = x - z * s is the formula to find the mean given the critical z-score and the critical raw score and the standard deviation.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "the critical raw score used in this formula is 400 and the standard deviation used is 2.2.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "when i used .841 as the critical z-score, this formula became:\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "m = 400 - .841 * 2.2 = 398.1498\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "when i used .842 as the critical z-score, this formula became:\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "m = 400 - .842 * 2.2 = 398.1476.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "when i used .84 as the critical z-score, this formula became:\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "m = 400 - .84 * 2.2 = 398.152.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "when z = .841, this is the result that i got:\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" src = \"http://theo.x10hosting.com/2018/082602.jpg\" alt=\"$$$\" >\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "when z = .842, this is the result that i got:\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" src = \"http://theo.x10hosting.com/2018/082603.jpg\" alt=\"$$$\" >\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "when z = .84, this is the result that i got:\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" src = \"http://theo.x10hosting.com/2018/082604.jpg\" alt=\"$$$\" >\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "you can see that:\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "a z-score of .841 gave me a mean of 398.1498 with an area of 20.02% to the right of the critical raw score of 400.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "a z-score of .842 gave me a mean 398.1476 with an area of 19.99% to the right of the critical raw score of 400.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "a z-score of .84 gave me a mean of 398.152 withy an area of 20.05% to the right of the critical raw score of 400.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "the z-score of .842 was the better z-score because it resulted in an area to the right of it as something slightly less than 20%.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "however, any one of these z-scores would have given me an answer that was well within 1% of the required area of 20% to the right of the critical z-score.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "the point is that you will get different answers depending on the method used to derive the critical z-score, but those answers will more then likely be well within a reasonably acceptable range of possible answers.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "since they want your answer to 2 decimal places, it appears that all of these will give you the same answer.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "that answer will be a mean of 398.15 rounded to 2 decimal places.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( " </SPAN>\n" );
document.write( "</TD></TR></TABLE>\n" );