document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1186596>Question 1186596</A>: <I><SPAN STYLE=\"word-break: break-all;\"> In a recent poll, 580 people were asked if they liked dogs, and 22% said they did. Find the margin of error of this poll, at the 95% confidence level.\r<BR>\n" );
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document.write( "Give your answer to three decimals<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 #849425 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=849425\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> Here's how to calculate the margin of error for this poll:\r<BR>\n" );
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document.write( "1. **Identify the key values:**\r<BR>\n" );
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document.write( "*   Sample size (n) = 580<BR>\n" );
document.write( "*   Sample proportion (p̂) = 0.22 (22% expressed as a decimal)<BR>\n" );
document.write( "*   Confidence level = 95%\r<BR>\n" );
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document.write( "2. **Find the critical z-score:**\r<BR>\n" );
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document.write( "*   For a 95% confidence level, the alpha (α) is 1 - 0.95 = 0.05.<BR>\n" );
document.write( "*   Since we are dealing with a two-tailed confidence interval, we divide alpha by 2: 0.05 / 2 = 0.025<BR>\n" );
document.write( "*   The z-score corresponding to 0.025 in each tail (or 0.975 in the center) is 1.96. You can find this using a z-table or calculator.\r<BR>\n" );
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document.write( "3. **Calculate the standard error:**\r<BR>\n" );
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document.write( "*   Standard Error (SE) = sqrt[ (p̂ * (1 - p̂)) / n ]<BR>\n" );
document.write( "*   SE = sqrt[ (0.22 * (1 - 0.22)) / 580 ]<BR>\n" );
document.write( "*   SE = sqrt[ (0.22 * 0.78) / 580 ]<BR>\n" );
document.write( "*   SE = sqrt(0.1716 / 580)<BR>\n" );
document.write( "*   SE ≈ sqrt(0.00029586)<BR>\n" );
document.write( "*   SE ≈ 0.0172\r<BR>\n" );
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document.write( "4. **Calculate the margin of error:**\r<BR>\n" );
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document.write( "*   Margin of Error (ME) = z * SE<BR>\n" );
document.write( "*   ME = 1.96 * 0.0172<BR>\n" );
document.write( "*   ME ≈ 0.0337\r<BR>\n" );
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document.write( "5. **Round to three decimals:**\r<BR>\n" );
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document.write( "*   ME ≈ 0.034\r<BR>\n" );
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document.write( "Therefore, the margin of error of this poll, at the 95% confidence level, is approximately 0.034.<BR>\n" );
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