document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1209947>Question 1209947</A>: <I><SPAN STYLE=\"word-break: break-all;\"> You intend to estimate a population mean with a confidence interval. You believe the population to have a normal distribution. Your sample size is 4.\r<BR>\n" );
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document.write( "While it is an uncommon confidence level, find the critical value that corresponds to a confidence level of 80.5%.<BR>\n" );
document.write( "(Report answer accurate to three decimal places with appropriate rounding.)\r<BR>\n" );
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document.write( "ta/2 +/- =  </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #851181 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=851181\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> Let $n$ be the sample size, which is $n=4$.<BR>\n" );
document.write( "The confidence level is 80.5%, or 0.805.<BR>\n" );
document.write( "The significance level $\alpha$ is $1 - 0.805 = 0.195$.<BR>\n" );
document.write( "The area in each tail is $\alpha/2 = 0.195/2 = 0.0975$.\r<BR>\n" );
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document.write( "Since the population is normally distributed and the sample size is small ($n=4$), we need to use a t-distribution with $n-1 = 4-1 = 3$ degrees of freedom.\r<BR>\n" );
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document.write( "We need to find the t-value $t_{\alpha/2}$ such that the area to the right of $t_{\alpha/2}$ is 0.0975.\r<BR>\n" );
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document.write( "We can use a t-table or a calculator to find this value.<BR>\n" );
document.write( "Using a t-table or calculator, we find that the t-value corresponding to an area of 0.0975 in the right tail with 3 degrees of freedom is approximately 1.589.\r<BR>\n" );
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document.write( "Therefore, $t_{\alpha/2} \approx 1.589$.\r<BR>\n" );
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document.write( "The critical value that corresponds to a confidence level of 80.5% is approximately 1.589.\r<BR>\n" );
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document.write( "Final Answer: The final answer is $\boxed{1.589}$<BR>\n" );
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