document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1170686>Question 1170686</A>: <I><SPAN STYLE=\"word-break: break-all;\"> In a survey, 25 people were asked how much they spent on their child's last birthday gift. The results were roughly bell-shaped with a mean of $43 and standard deviation of $8. Find the margin of error at a 80% confidence level.<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 #851117 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=851117\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> Let's calculate the margin of error for this problem.\r<BR>\n" );
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document.write( "**Given:**\r<BR>\n" );
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document.write( "* Sample size (n) = 25<BR>\n" );
document.write( "* Sample mean (x̄) = $43<BR>\n" );
document.write( "* Sample standard deviation (s) = $8<BR>\n" );
document.write( "* Confidence level = 80%\r<BR>\n" );
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document.write( "**1. Find the Critical t-value (t_c)**\r<BR>\n" );
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document.write( "Since the sample size is small (n = 25) and the population standard deviation is unknown, we will use the t-distribution.\r<BR>\n" );
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document.write( "* Degrees of freedom (df) = n - 1 = 25 - 1 = 24<BR>\n" );
document.write( "* Confidence level = 80%, so α = 1 - 0.80 = 0.20<BR>\n" );
document.write( "* Since it's a two-tailed test, α/2 = 0.20 / 2 = 0.10\r<BR>\n" );
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document.write( "Now, we need to find the t-value from the t-distribution table or using a calculator with df = 24 and α/2 = 0.10.\r<BR>\n" );
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document.write( "* t_c ≈ 1.318\r<BR>\n" );
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document.write( "**2. Calculate the Margin of Error (E)**\r<BR>\n" );
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document.write( "The formula for the margin of error is:\r<BR>\n" );
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document.write( "E = t_c * (s / √n)\r<BR>\n" );
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document.write( "Plug in the values:\r<BR>\n" );
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document.write( "E = 1.318 * (8 / √25)<BR>\n" );
document.write( "E = 1.318 * (8 / 5)<BR>\n" );
document.write( "E = 1.318 * 1.6<BR>\n" );
document.write( "E = 2.1088\r<BR>\n" );
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document.write( "**3. Round to a Reasonable Number of Decimal Places**\r<BR>\n" );
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document.write( "Let's round to two decimal places, since we're dealing with money:\r<BR>\n" );
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document.write( "E ≈ $2.11\r<BR>\n" );
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document.write( "**Answer:**\r<BR>\n" );
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document.write( "The margin of error at an 80% confidence level is approximately $2.11.<BR>\n" );
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