document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1181343>Question 1181343</A>: <I><SPAN STYLE=\"word-break: break-all;\"> In a study of 1003 community college students, 521 indicated that they believe television is a luxury that they can live without.\r<BR>\n" );
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document.write( "    a) (15 points) Construct a 95% confidence interval for the population proportion of community college students who believe television is a luxury that they can live without.\r<BR>\n" );
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document.write( "    b) (5 points) Explain what the confidence interval you constructed means. </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #850062 by </B><A HREF=/tutors/aboutme.mpl?userid=CPhill><B>CPhill(2189)</B></A><img src=\"/images/stars/stars-13.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=850062\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> Here's how to construct and interpret the confidence interval:\r<BR>\n" );
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document.write( "**a) Constructing a 95% Confidence Interval:**\r<BR>\n" );
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document.write( "1. **Calculate the sample proportion (p̂):**\r<BR>\n" );
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document.write( "p̂ = (Number of students who believe TV is a luxury) / (Total number of students)<BR>\n" );
document.write( "p̂ = 521 / 1003<BR>\n" );
document.write( "p̂ ≈ 0.52\r<BR>\n" );
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document.write( "2. **Find the critical z-value:**\r<BR>\n" );
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document.write( "For a 95% confidence level, the alpha (α) level is 1 - 0.95 = 0.05.  We need to find the z-score that corresponds to an area of 0.975 (1 - α/2) in the standard normal distribution.  This z-value is approximately 1.96. You can find this using a z-table or calculator.\r<BR>\n" );
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document.write( "3. **Calculate the margin of error (E):**\r<BR>\n" );
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document.write( "E = z * √(p̂(1 - p̂) / n)<BR>\n" );
document.write( "E = 1.96 * √(0.52 * (1 - 0.52) / 1003)<BR>\n" );
document.write( "E ≈ 1.96 * √(0.2496 / 1003)<BR>\n" );
document.write( "E ≈ 1.96 * 0.0157<BR>\n" );
document.write( "E ≈ 0.0308\r<BR>\n" );
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document.write( "4. **Construct the confidence interval:**\r<BR>\n" );
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document.write( "Lower Bound = p̂ - E = 0.52 - 0.0308 ≈ 0.4892<BR>\n" );
document.write( "Upper Bound = p̂ + E = 0.52 + 0.0308 ≈ 0.5508\r<BR>\n" );
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document.write( "Therefore, the 95% confidence interval is approximately (0.4892, 0.5508).\r<BR>\n" );
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document.write( "**b) Interpretation of the Confidence Interval:**\r<BR>\n" );
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document.write( "We are 95% confident that the true population proportion of community college students who believe television is a luxury they can live without is between 48.92% and 55.08%.  This means that if we were to repeat this study many times and calculate a 95% confidence interval each time, 95% of those intervals would contain the true population proportion.  It's important to remember that this interval is an estimate, and the true proportion may or may not fall within this range.<BR>\n" );
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