document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1165661>Question 1165661</A>: <I><SPAN STYLE=\"word-break: break-all;\"> A college statistics professor has office hours from 9:00 am to 10:30 am daily. A random sample of 20 students waiting to see the professor was selected to estimate the mean waiting time for all students. This sample produced a 95% confidence interval of 18.5 to 26.1 minutes. How should the professor interpret this interval?\r<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 #852950 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=852950\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> The professor should interpret the $95\%$ confidence interval of $18.5$ to $26.1$ minutes as follows:\r<BR>\n" );
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document.write( "## 1. Primary Interpretation\r<BR>\n" );
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document.write( "The interval is an estimate of the true average waiting time ($\mu$) for **all students** who visit during office hours.\r<BR>\n" );
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document.write( "The correct interpretation is: **The professor can be 95% confident that the true mean waiting time ($\mu$) for all students is between $18.5$ minutes and $26.1$ minutes.**\r<BR>\n" );
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document.write( "## 2. Interpretation of Confidence Level\r<BR>\n" );
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document.write( "The $95\%$ confidence level refers to the **method** used to construct the interval, not the interval itself.\r<BR>\n" );
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document.write( "* If the professor were to take **many** random samples of 20 students and construct a $95\%$ confidence interval from each sample, approximately **95% of those intervals would capture the true mean waiting time** ($\mu$).<BR>\n" );
document.write( "* Conversely, about $5\%$ of the intervals constructed would fail to capture the true mean.\r<BR>\n" );
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document.write( "## 3. What the Interval **DOES NOT** Mean\r<BR>\n" );
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document.write( "The professor should be careful to avoid common misinterpretations:\r<BR>\n" );
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document.write( "* **It does not mean** that $95\%$ of all students wait between $18.5$ and $26.1$ minutes. (The interval estimates the **mean**, not individual waiting times.)<BR>\n" );
document.write( "* **It does not mean** there is a $95\%$ chance the true mean is exactly $22.3$ minutes (the midpoint). The true mean is a fixed value; it is either in the interval or it is not. </SPAN>\n" );
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