document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1165774>Question 1165774</A>: <I><SPAN STYLE=\"word-break: break-all;\"> A manufacturing process with acceptance limits 100+_3 A recent sample produces measurements in file DATA.Assuming normal distribution. What percentage of production does not meet the acceptance limits?<BR>\n" );
document.write( "Answer <BR>\n" );
document.write( "a) 0.81%, b) 0.14%, c) 0.46%, d) 0.67% </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #852936 by </B><A HREF=/tutors/aboutme.mpl?userid=CPhill><B>CPhill(2189)</B></A><img src=\"/images/stars/stars-13.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=852936\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> ```python?code_reference&code_event_index=2<BR>\n" );
document.write( "import os\r<BR>\n" );
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document.write( "print(os.listdir())\r<BR>\n" );
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document.write( "```\r<BR>\n" );
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document.write( "```text?code_stdout&code_event_index=2<BR>\n" );
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document.write( "```\r<BR>\n" );
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document.write( "I cannot calculate the percentage without the data from the file named **DATA**.\r<BR>\n" );
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document.write( "The file seems to be missing from the current working directory. Please upload the file (e.g., as `DATA.csv`) or provide the **sample mean ($\bar{x}$) and sample standard deviation ($s$)** of the measurements, which are needed to model the normal distribution.\r<BR>\n" );
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document.write( "### General Steps for the Calculation (Once Data is Provided)\r<BR>\n" );
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document.write( "Assuming a normal distribution with mean $\mu$ and standard deviation $\sigma$ derived from your sample, the process is:\r<BR>\n" );
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document.write( "1.  **Define Limits:**<BR>\n" );
document.write( "      * Lower Specification Limit (LSL) $= 100 - 3 = 97$<BR>\n" );
document.write( "      * Upper Specification Limit (USL) $= 100 + 3 = 103$<BR>\n" );
document.write( "2.  **Calculate Z-scores:**<BR>\n" );
document.write( "    $$Z_{LSL} = \frac{\text{LSL} - \mu}{\sigma} \quad \text{and} \quad Z_{USL} = \frac{\text{USL} - \mu}{\sigma}$$<BR>\n" );
document.write( "3.  **Find the Percentage Out of Spec:**<BR>\n" );
document.write( "    The total percentage not meeting the limits is:<BR>\n" );
document.write( "    $$P(\text{Defective}) = P(X < 97) + P(X > 103)$$\r<BR>\n" );
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document.write( "Once you provide the data, I can complete the calculation and match the result to one of the options. </SPAN>\n" );
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