document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1209556>Question 1209556</A>: <I><SPAN STYLE=\"word-break: break-all;\"> The length of a pencil is supposed to fit within the tolerance 7.25 ± 0.05 in. However, it has been formally distributed with a tolerance of 7.2 ± 0.02 in. However, due to new machines, the tolerance has increased to 7.25 ± 0.025 in. The mean has remained constant. What is the tolerance used to produce the same percentage of pencils before the increase? </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #849419 by </B><A HREF=/tutors/aboutme.mpl?userid=CPhill><B>CPhill(1987)</B></A><img src=\"/images/stars/stars-12.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=CPhill><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=CPhill><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=849419\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> Here's how to solve this problem:\r<BR>\n" );
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document.write( "**Understanding the Concept**\r<BR>\n" );
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document.write( "The tolerance represents the acceptable variation in pencil length.  A smaller tolerance means more consistent lengths, but also potentially more rejects.  We're looking for the tolerance that, when combined with the original mean, would result in the same *proportion* of acceptable pencils as the *new* tolerance.\r<BR>\n" );
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document.write( "**Steps**\r<BR>\n" );
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document.write( "1. **Determine the Z-score:**  The tolerance is related to the standard deviation (a measure of how spread out the lengths are) and a Z-score.  The Z-score tells us how many standard deviations away from the mean a particular length is.  We need to find the Z-score that corresponds to the desired percentage of pencils within the tolerance.  Since the problem implies we are looking at a 6-sigma process, we can assume that the percentage of pencils within the tolerance is 99.73%\r<BR>\n" );
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document.write( "2. **Calculate the Original Standard Deviation:** We can use the original tolerance to calculate the standard deviation of the process, using the Z-score we calculated above.\r<BR>\n" );
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document.write( "3. **Calculate the new standard deviation:** We can use the new tolerance to calculate the standard deviation of the process, using the Z-score we calculated above.\r<BR>\n" );
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document.write( "4. **Calculate the Equivalent Tolerance:** Now that we have the standard deviation of the new distribution, we can calculate the tolerance that would produce the same percentage of pencils under the original mean, using the Z-score we calculated above.\r<BR>\n" );
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document.write( "**Calculation**\r<BR>\n" );
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document.write( "1. **Z-score:** For a 99.73% confidence interval, the Z-score is approximately 3.\r<BR>\n" );
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document.write( "2. **Original Standard Deviation:** original_std = original_tolerance / 3\r<BR>\n" );
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document.write( "3. **New Standard Deviation:** new_std = new_tolerance / 3\r<BR>\n" );
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document.write( "4. **Equivalent Tolerance:** equivalent_tolerance = 3 * original_std\r<BR>\n" );
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document.write( "**Result**\r<BR>\n" );
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document.write( "The tolerance used to produce the same percentage of pencils before the increase is 0.05 in.<BR>\n" );
document.write( " </SPAN>\n" );
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