document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1172781>Question 1172781</A>: <I><SPAN STYLE=\"word-break: break-all;\"> What should the control limits be placed in the design of the process sets alpha = 0.01 with the following parameters (assume that the sample size condition has been verified)?<BR>\n" );
document.write( "(a) mu.png= 100,sigma.png= 20, and n = 25 cases per batch\r<BR>\n" );
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document.write( "(b) mu.png= 2000,sigma.png= 2000, and n = 100 cases per batch<BR>\n" );
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document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #850805 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=850805\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> To determine the control limits for an X-bar chart, we need to calculate the upper control limit (UCL) and the lower control limit (LCL). We'll use the following formula:\r<BR>\n" );
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document.write( "* UCL = μ + Z(α/2) * (σ / √n)<BR>\n" );
document.write( "* LCL = μ - Z(α/2) * (σ / √n)\r<BR>\n" );
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document.write( "Where:\r<BR>\n" );
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document.write( "* μ = population mean<BR>\n" );
document.write( "* σ = population standard deviation<BR>\n" );
document.write( "* n = sample size<BR>\n" );
document.write( "* Z(α/2) = the Z-score corresponding to the desired significance level (alpha).\r<BR>\n" );
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document.write( "Given that alpha = 0.01, we need to find the Z-score for alpha/2 = 0.005. This means we are looking for the Z-score that leaves 0.005 in the upper tail of the standard normal distribution. Using a Z-table or calculator, we find that Z(0.005) ≈ 2.576.\r<BR>\n" );
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document.write( "**a. μ = 100, σ = 20, n = 25**\r<BR>\n" );
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document.write( "1.  **Calculate the standard error of the mean (SEM):**<BR>\n" );
document.write( "    * SEM = σ / √n = 20 / √25 = 20 / 5 = 4\r<BR>\n" );
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document.write( "2.  **Calculate the control limits:**<BR>\n" );
document.write( "    * UCL = 100 + 2.576 * 4 = 100 + 10.304 ≈ 110.30<BR>\n" );
document.write( "    * LCL = 100 - 2.576 * 4 = 100 - 10.304 ≈ 89.70\r<BR>\n" );
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document.write( "Therefore, the control limits for (a) are approximately 89.70 and 110.30.\r<BR>\n" );
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document.write( "**b. μ = 2000, σ = 2000, n = 100**\r<BR>\n" );
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document.write( "1.  **Calculate the standard error of the mean (SEM):**<BR>\n" );
document.write( "    * SEM = σ / √n = 2000 / √100 = 2000 / 10 = 200\r<BR>\n" );
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document.write( "2.  **Calculate the control limits:**<BR>\n" );
document.write( "    * UCL = 2000 + 2.576 * 200 = 2000 + 515.2 ≈ 2515.2<BR>\n" );
document.write( "    * LCL = 2000 - 2.576 * 200 = 2000 - 515.2 ≈ 1484.8\r<BR>\n" );
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document.write( "Therefore, the control limits for (b) are approximately 1484.8 and 2515.2.<BR>\n" );
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