document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1182683>Question 1182683</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Assume that we want to construct a confidence interval. Do one of the​ following, as​ appropriate: (a) find the critical value tα/2​, ​(b) find the critical value zα/2​, or​ (c) state that neither the normal distribution nor the t distribution applies.\r<BR>\n" );
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document.write( "The confidence level is 95​%, σ is not​ known, and the normal quantile plot of the 17 salaries​ (in thousands of​ dollars) of basketball players on a team is as shown.\r<BR>\n" );
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document.write( "Select the correct choice below​ and, if​ necessary, fill in the blank to complete your choice.<BR>\n" );
document.write( "A. tα/2 = __<BR>\n" );
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document.write( "B. zα/2 = __<BR>\n" );
document.write( "(Round to two decimal places as​ needed.)\r<BR>\n" );
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document.write( "C. Neither the normal distribution nor the t distribution applies. </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #849910 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=849910\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> The correct choice is **A. tα/2 = __**\r<BR>\n" );
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document.write( "Here's why:\r<BR>\n" );
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document.write( "1. **σ is not known:** When the population standard deviation (σ) is unknown, we use the t-distribution to construct confidence intervals.\r<BR>\n" );
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document.write( "2. **Normal quantile plot:** The normal quantile plot helps us assess if the sample data comes from a normally distributed population. If the data is approximately normal, then the t-distribution is appropriate. A roughly linear pattern on the normal quantile plot suggests that the data are approximately normally distributed. Since we have a normal quantile plot, we can assume that the underlying population is normally distributed.\r<BR>\n" );
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document.write( "3. **Calculating tα/2:**<BR>\n" );
document.write( "   * Confidence level = 95% = 0.95<BR>\n" );
document.write( "   * α = 1 - Confidence level = 1 - 0.95 = 0.05<BR>\n" );
document.write( "   * α/2 = 0.05 / 2 = 0.025<BR>\n" );
document.write( "   * Degrees of freedom (df) = n - 1 = 17 - 1 = 16\r<BR>\n" );
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document.write( "Now, look up the t-value corresponding to α/2 = 0.025 and df = 16 in a t-table or use a calculator. You should find a value of approximately 2.12.\r<BR>\n" );
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document.write( "Therefore, tα/2 = **2.12**<BR>\n" );
document.write( " </SPAN>\n" );
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