document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1208817>Question 1208817</A>: <I><SPAN STYLE=\"word-break: break-all;\"> You plan to conduct a survey to find what proportion of the workforce has two or more jobs. You decide on the 99% confidence level and a margin of error of 2%. A pilot survey reveals that 8 of the 47 sampled hold two or more jobs. (Use t Distribution Table & z Distribution Table.)\r<BR>\n" );
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document.write( "How many in the workforce should be interviewed to meet your requirements? (Round z-score to 2 decimal places. Round up your answer to the next whole number.)<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 #849185 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=849185\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> Here's how to calculate the necessary sample size:\r<BR>\n" );
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document.write( "**1. Identify Key Values:**\r<BR>\n" );
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document.write( "* **Confidence Level:** 99%<BR>\n" );
document.write( "* **Margin of Error (E):** 2% = 0.02<BR>\n" );
document.write( "* **Pilot Survey:** 8 out of 47 have two or more jobs.  This gives us a preliminary estimate of the proportion (p̂):  p̂ = 8/47 ≈ 0.1702\r<BR>\n" );
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document.write( "**2. Find the Z-score:**\r<BR>\n" );
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document.write( "For a 99% confidence level, the alpha (α) is 1 - 0.99 = 0.01.  Since it's a two-tailed test, we divide alpha by 2: 0.01 / 2 = 0.005.  We want the z-score that corresponds to an area of 0.995 (1 - 0.005) in the standard normal distribution table.\r<BR>\n" );
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document.write( "Z-score ≈ 2.58 (rounded to two decimal places).\r<BR>\n" );
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document.write( "**3. Use the Sample Size Formula:**\r<BR>\n" );
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document.write( "Since we have a preliminary estimate of the proportion from the pilot study, we use the following formula for sample size:\r<BR>\n" );
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document.write( "n = (Z² * p̂ * (1 - p̂)) / E²\r<BR>\n" );
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document.write( "Where:\r<BR>\n" );
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document.write( "* n = sample size<BR>\n" );
document.write( "* Z = Z-score<BR>\n" );
document.write( "* p̂ = estimated proportion<BR>\n" );
document.write( "* E = margin of error\r<BR>\n" );
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document.write( "**4. Plug in the Values and Calculate:**\r<BR>\n" );
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document.write( "n = (2.58² * 0.1702 * (1 - 0.1702)) / 0.02²<BR>\n" );
document.write( "n = (6.6564 * 0.1702 * 0.8298) / 0.0004<BR>\n" );
document.write( "n = 0.9457 / 0.0004<BR>\n" );
document.write( "n ≈ 2364.25\r<BR>\n" );
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document.write( "**5. Round Up:**\r<BR>\n" );
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document.write( "Always round the sample size *up* to the next whole number to ensure the desired confidence level and margin of error are met.\r<BR>\n" );
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document.write( "n = 2365\r<BR>\n" );
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document.write( "**Answer:**\r<BR>\n" );
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document.write( "You should interview 2365 people in the workforce to meet your requirements.<BR>\n" );
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