document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1166449>Question 1166449</A>: <I><SPAN STYLE=\"word-break: break-all;\"> in a made up poll the proportion of people who like dark chocolate more than milk chocolate was 43% with a margin of error of 1.9% describe the conclusion of p(proportion) using an absolute value inequality using the </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #852755 by </B><A HREF=/tutors/aboutme.mpl?userid=CPhill><B>CPhill(2138)</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=852755\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> The conclusion of the poll regarding the proportion ($p$) of people who prefer dark chocolate over milk chocolate can be described by the following absolute value inequality:\r<BR>\n" );
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document.write( "$$|p - 0.43| \leq 0.019$$\r<BR>\n" );
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document.write( "---\r<BR>\n" );
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document.write( "## Explanation\r<BR>\n" );
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document.write( "This inequality expresses that the true proportion ($p$) is estimated to be $0.43$, with a maximum possible difference (or error) of $0.019$ in either direction.\r<BR>\n" );
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document.write( "1.  **Point Estimate:** The sample proportion is $43\%$, or **$0.43$**. This is the center of the confidence interval.<BR>\n" );
document.write( "2.  **Margin of Error ($E$):** The error is $1.9\%$, or **$0.019$**.<BR>\n" );
document.write( "3.  **Absolute Value Inequality:** The standard form for a confidence interval's conclusion is:<BR>\n" );
document.write( "    $$|\text{True Proportion} - \text{Sample Proportion}| \leq \text{Margin of Error}$$\r<BR>\n" );
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document.write( "### Confidence Interval (Expanded Form)\r<BR>\n" );
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document.write( "The inequality can be rewritten as a compound inequality:<BR>\n" );
document.write( "$$-0.019 \leq p - 0.43 \leq 0.019$$\r<BR>\n" );
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document.write( "Adding $0.43$ to all parts gives the confidence interval:<BR>\n" );
document.write( "$$0.43 - 0.019 \leq p \leq 0.43 + 0.019$$<BR>\n" );
document.write( "$$0.411 \leq p \leq 0.449$$\r<BR>\n" );
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document.write( "The poll concludes that the true proportion of people who prefer dark chocolate is between **$41.1\%$ and $44.9\%$**. </SPAN>\n" );
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