document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1192336>Question 1192336</A>: <I><SPAN STYLE=\"word-break: break-all;\"> A manufacturer wishes to estimate the proportion of washing machines leaving the factory that is defective. How large a sample should she check in order to be 94 percent confident that the true proportion is estimated to within 0.015?\r<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 #849042 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=849042\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> **1. Determine the z-score:**\r<BR>\n" );
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document.write( "* Find the z-score that corresponds to a 94% confidence level. This means we want to find the z-score that leaves 3% in each tail of the standard normal distribution.<BR>\n" );
document.write( "* Using a standard normal distribution table or calculator, we find that z ≈ 1.8808.\r<BR>\n" );
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document.write( "**2. Estimate the proportion of defective washing machines:**\r<BR>\n" );
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document.write( "* Since we don't have any prior information about the proportion of defective washing machines, we'll use the most conservative estimate: p = 0.5. This will give us the largest sample size needed.\r<BR>\n" );
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document.write( "**3. Calculate the sample size:**\r<BR>\n" );
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document.write( "* Use the following formula for determining the sample size needed for estimating a population proportion:\r<BR>\n" );
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document.write( "   n = (z² * p * (1 - p)) / E² \r<BR>\n" );
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document.write( "   where:<BR>\n" );
document.write( "      * n is the sample size<BR>\n" );
document.write( "      * z is the z-score <BR>\n" );
document.write( "      * p is the estimated proportion of defective washing machines<BR>\n" );
document.write( "      * E is the desired margin of error (0.015 in this case)\r<BR>\n" );
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document.write( "* Plug in the values:\r<BR>\n" );
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document.write( "   n = (1.8808² * 0.5 * (1 - 0.5)) / 0.015²<BR>\n" );
document.write( "   n ≈ 7861.16\r<BR>\n" );
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document.write( "* **Round up:** Since we can't have a fraction of a washing machine, we round up to the nearest whole number.\r<BR>\n" );
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document.write( "**Therefore, the manufacturer should check a sample of 7862 washing machines to be 94% confident that the true proportion of defective washing machines is estimated to within 0.015.**<BR>\n" );
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