document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1162169>Question 1162169</A>: <I><SPAN STYLE=\"word-break: break-all;\"> a machine is packaging nominal weight is 800 grams packets and it has been found that over a long period the actual weights put in the packets has been normally distributed with a standard deviation of 5 grams .a company wishes to ensure that no more than 1% of the packets has the weight of less than 800 <BR>\n" );
document.write( "grams ;at what mean should a machine be set? <BR>\n" );
document.write( "the weekly outputs of packets average 2,400,000 and the cost in pence,of the producing a packet of weight w gram is given by relationship c=6+0.5w<BR>\n" );
document.write( "if installation anew machinery , the standard deviation could be reduced to 2 grams,and the mean allowed to fall just far enough to give the same percentage bellow 800 grams as before , determine the average saving in dollars pr week </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #785904 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=785904\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> the critical z-score is -2.326347877.<BR>\n" );
document.write( "the area to the left of this z-score will be .01, which is 1%.\r<BR>\n" );
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document.write( "you want less than or equal to 1% of all the packets made to be equal to 800.\r<BR>\n" );
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document.write( "the z-score formula is z = (x - m) / x<BR>\n" );
document.write( "z is the z-score<BR>\n" );
document.write( "x is the raw score<BR>\n" );
document.write( "m is the raw mean<BR>\n" );
document.write( "s is the standard deviation.\r<BR>\n" );
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document.write( "in this formula, z = -2.326347877, s = 5, and x = 800.<BR>\n" );
document.write( "you want to solve for m.\r<BR>\n" );
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document.write( "formula becomes -2.326347877 = (800 - m) / 5.<BR>\n" );
document.write( "multiply both sides of the equation by 5 to get:<BR>\n" );
document.write( "-2.326347877 * 5 = 800 - m<BR>\n" );
document.write( "add m to both sides of the equation and subtract -2.326347877 * 5 from both sides of the equation to get:<BR>\n" );
document.write( "m = 800 - (-2.326347877 * 5) = 811.6317394.\r<BR>\n" );
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document.write( "that's what the mean needs to be so that less than or equal to 1% of the packet weighs less than 800 when the standard deviation is 5.\r<BR>\n" );
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document.write( "if the standard deviation is equal to 2, then the formula to find m becomes:<BR>\n" );
document.write( "-2.326347877 = (800 - m) / 2.<BR>\n" );
document.write( "solve for m as before to get:<BR>\n" );
document.write( "m = 800 - (-2.326347877 * 2) = 804.6526958\r<BR>\n" );
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document.write( "that's what the mean needs to be so that less than or equal to 1% of the packet weighs less than 800 when the standard deviation is 2.\r<BR>\n" );
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document.write( "an average of 2,400,000 packets are produced each week.\r<BR>\n" );
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document.write( "when the mean is 811.6317394, the cost is 2,400,000 * (6 + .5 * 811.6317394) = 988,358,087.3 dollars.\r<BR>\n" );
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document.write( "when the mean is 804.6526958, the cost is 2,400,000 * (6 + .5 * 804.6526958) = 979,983,234.9 dollars.\r<BR>\n" );
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document.write( "the savings are equal to 988,358,087.3 dollars minus 979,983,234.9 dollars = 8,374,852.358 dollars.\r<BR>\n" );
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document.write( "that's 8.3754852358 million dollars, which can also be shown as 8.3754852358 * 10^6 dollars.\r<BR>\n" );
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document.write( "i used statistical calculators to confirm the manual calculations.\r<BR>\n" );
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document.write( "here's what they showed.\r<BR>\n" );
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document.write( "they confirmed the manual calculations were correct.\r<BR>\n" );
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document.write( "in both cases, the number of packets weighing less than 800 grams was less than or equal to 1%.\r<BR>\n" );
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document.write( "the calculators can be found at:\r<BR>\n" );
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document.write( "<a  rel=nofollow HREF= \"https://www.omnicalculator.com/statistics/normal-distribution\" target = \"_blank\">https://www.omnicalculator.com/statistics/normal-distribution</a>\r<BR>\n" );
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document.write( "<a  rel=nofollow HREF= \"http://davidmlane.com/hyperstat/z_table.html\" target = \"_blank\">http://davidmlane.com/hyperstat/z_table.html</a> </SPAN>\n" );
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