document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1178026>Question 1178026</A>: <I><SPAN STYLE=\"word-break: break-all;\"> An airline has surveyed a simple random sample of air travelers to find out whether they would be interested in paying a higher fare in order to have access to email during their flight:Of the 400 travelers surveyed, 80 said email access would be worth a slight extra cost. Construct a 95% confidence interval for the population proportion of air travelers to who are in favour of the airline's email idea.  </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #850376 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=850376\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> Let's construct the 95% confidence interval for the population proportion of air travelers who are in favor of the airline's email idea.\r<BR>\n" );
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document.write( "**1. Calculate the Sample Proportion (p̂)**\r<BR>\n" );
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document.write( "* **Sample Size (n):** 400<BR>\n" );
document.write( "* **Number in Favor (x):** 80<BR>\n" );
document.write( "* **Sample Proportion (p̂):** p̂ = x / n = 80 / 400 = 0.20\r<BR>\n" );
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document.write( "**2. Calculate the Standard Error (SE)**\r<BR>\n" );
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document.write( "* **Standard Error (SE):** SE = √[p̂(1 - p̂) / n]<BR>\n" );
document.write( "* SE = √[0.20(1 - 0.20) / 400]<BR>\n" );
document.write( "* SE = √[0.20(0.80) / 400]<BR>\n" );
document.write( "* SE = √(0.16 / 400)<BR>\n" );
document.write( "* SE = √0.0004 = 0.02\r<BR>\n" );
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document.write( "**3. Find the Critical Value (z*)**\r<BR>\n" );
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document.write( "* **Confidence Level:** 95%<BR>\n" );
document.write( "* **Alpha (α):** 1 - 0.95 = 0.05<BR>\n" );
document.write( "* **Alpha/2:** α/2 = 0.025<BR>\n" );
document.write( "* **Critical Value (z*):** For a 95% confidence interval, z* = 1.96 (from the standard normal distribution table)\r<BR>\n" );
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document.write( "**4. Calculate the Margin of Error (ME)**\r<BR>\n" );
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document.write( "* **Margin of Error (ME):** ME = z* * SE<BR>\n" );
document.write( "* ME = 1.96 * 0.02 = 0.0392\r<BR>\n" );
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document.write( "**5. Construct the Confidence Interval**\r<BR>\n" );
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document.write( "* **Confidence Interval:** p̂ ± ME<BR>\n" );
document.write( "* **Lower Bound:** p̂ - ME = 0.20 - 0.0392 = 0.1608<BR>\n" );
document.write( "* **Upper Bound:** p̂ + ME = 0.20 + 0.0392 = 0.2392\r<BR>\n" );
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document.write( "**Therefore, the 95% confidence interval for the population proportion of air travelers who are in favor of the airline's email idea is (0.1608, 0.2392).**<BR>\n" );
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