document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1204322>Question 1204322</A>: <I><SPAN STYLE=\"word-break: break-all;\"> The expected number of defective parts produced on an assembly line per shift is 50 with a standard deviation of 8. Use Chebyshev's inequality to find the minimum probability that the number of defective parts on a particular shift will be between 22 and 78. (Round your answer to four decimal places.) </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #848064 by </B><A HREF=/tutors/aboutme.mpl?userid=GingerAle><B>GingerAle(43)</B></A><img src=\"/images/stars/stars-07.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=GingerAle><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/images/nopicture.jpg><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=848064\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> **1. Define the Range**\r<BR>\n" );
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document.write( "* We are interested in the range of defective parts between 22 and 78. <BR>\n" );
document.write( "* Mean (μ) = 50 <BR>\n" );
document.write( "* Standard Deviation (σ) = 8\r<BR>\n" );
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document.write( "**2. Calculate the Number of Standard Deviations (k)**\r<BR>\n" );
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document.write( "* **Lower Bound:** (50 - 22) / 8 = 3.5 standard deviations below the mean<BR>\n" );
document.write( "* **Upper Bound:** (78 - 50) / 8 = 3.5 standard deviations above the mean\r<BR>\n" );
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document.write( "**3. Apply Chebyshev's Inequality**\r<BR>\n" );
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document.write( "* Chebyshev's Inequality states that for any data set, at least 1 - (1/k²) of the data values will fall within k standard deviations of the mean.\r<BR>\n" );
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document.write( "* In this case, k = 3.5\r<BR>\n" );
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document.write( "* Probability (22 ≤ Defective Parts ≤ 78) ≥ 1 - (1/3.5²) <BR>\n" );
document.write( "                                     ≥ 1 - (1/12.25) <BR>\n" );
document.write( "                                     ≥ 0.9184\r<BR>\n" );
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document.write( "**Therefore, according to Chebyshev's Inequality, the minimum probability that the number of defective parts on a particular shift will be between 22 and 78 is 0.9184.**\r<BR>\n" );
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document.write( "**Note:**\r<BR>\n" );
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document.write( "* Chebyshev's Inequality provides a lower bound for the probability. The actual probability may be higher.<BR>\n" );
document.write( "* This inequality is applicable to any distribution, regardless of its shape. \r<BR>\n" );
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document.write( "Let me know if you have any other questions! <BR>\n" );
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