document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1193442>Question 1193442</A>: <I><SPAN STYLE=\"word-break: break-all;\"> A shipment of 10 items has two defective and eight non defective units. In the inspection of the shipment , a sample of units will be selected and tested. If a defective unit is found , the shipment of 10 units will be rejected.If management would like a .90 probability of rejecting a shipment with two defective and eight non defective units , how large a sample would you recommend ? </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #848805 by </B><A HREF=/tutors/aboutme.mpl?userid=CPhill><B>CPhill(1959)</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=848805\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> **1. Define the Probability of Rejection:**\r<BR>\n" );
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document.write( "* The probability of rejecting a shipment is the probability of finding at least one defective unit in the sample.\r<BR>\n" );
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document.write( "**2. Calculate the Probability of Not Rejecting:**\r<BR>\n" );
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document.write( "* The probability of not rejecting a shipment is the probability of finding no defective units in the sample.\r<BR>\n" );
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document.write( "**3. Determine the Sample Size:**\r<BR>\n" );
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document.write( "* We need to find the sample size (n) that satisfies the following condition:<BR>\n" );
document.write( "   Probability of rejecting the shipment ≥ 0.90<BR>\n" );
document.write( "   1 - Probability of not rejecting the shipment ≥ 0.90<BR>\n" );
document.write( "   Probability of not rejecting the shipment ≤ 0.10\r<BR>\n" );
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document.write( "**4. Calculate the Probability of Not Rejecting (for a given sample size n):**\r<BR>\n" );
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document.write( "* The probability of not rejecting a shipment is the probability of selecting only non-defective units in the sample.<BR>\n" );
document.write( "* This can be calculated using the hypergeometric distribution:\r<BR>\n" );
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document.write( "   Probability of not rejecting = (Number of ways to choose n non-defective units from 8 non-defective units) / (Total number of ways to choose n units from 10 total units)<BR>\n" );
document.write( "   Probability of not rejecting = (8Cn) / (10Cn) \r<BR>\n" );
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document.write( "**5. Find the Sample Size (n) by Trial and Error:**\r<BR>\n" );
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document.write( "* Start with a small sample size (e.g., n = 1) and calculate the probability of not rejecting.<BR>\n" );
document.write( "* Increase the sample size gradually and recalculate the probability until the probability of not rejecting is less than or equal to 0.10.\r<BR>\n" );
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document.write( "**Using a Spreadsheet or Statistical Software:**\r<BR>\n" );
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document.write( "* You can use a spreadsheet program (like Excel) or statistical software (like R or Python) to efficiently calculate the hypergeometric probabilities for different sample sizes. This will help you quickly find the minimum sample size that meets the desired rejection probability.\r<BR>\n" );
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document.write( "**Example (Illustrative):**\r<BR>\n" );
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document.write( "* **n = 1:** Probability of not rejecting = (8C1) / (10C1) = 0.8 <BR>\n" );
document.write( "* **n = 2:** Probability of not rejecting = (8C2) / (10C2) = 0.56<BR>\n" );
document.write( "* **n = 3:** Probability of not rejecting = (8C3) / (10C3) = 0.3 <BR>\n" );
document.write( "* **n = 4:** Probability of not rejecting = (8C4) / (10C4) = 0.1429 <BR>\n" );
document.write( "* **n = 5:** Probability of not rejecting = (8C5) / (10C5) = 0.051 \r<BR>\n" );
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document.write( "In this example, a sample size of **n = 5** would likely satisfy the requirement, as the probability of not rejecting is less than 0.10.\r<BR>\n" );
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document.write( "**Note:**\r<BR>\n" );
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document.write( "* This is a simplified example. In real-world scenarios, you might need to consider factors like the cost of inspection, the potential consequences of rejecting a good shipment, and the desired level of consumer confidence.<BR>\n" );
document.write( "* Statistical software or tables can provide more accurate and efficient calculations for the hypergeometric distribution.\r<BR>\n" );
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document.write( "I hope this helps! Let me know if you have any further questions.<BR>\n" );
document.write( " </SPAN>\n" );
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