document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1191528>Question 1191528</A>: <I><SPAN STYLE=\"word-break: break-all;\"> A population consists of thE following values:12, 12, 14, and 16. A. List all samples of size 2, and compute the mean of each sample. B. Compute the mean of the distribution of sample means and the population mean. Compare the two values<BR>\n" );
document.write( "c. Compare the dispersion in the population with that of the sample means. </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #849169 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=849169\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> Here's how to analyze the given population and its samples:\r<BR>\n" );
document.write( "\n" );
document.write( "**A. Listing Samples and Computing Means:**\r<BR>\n" );
document.write( "\n" );
document.write( "Since the population is small, we can list all possible samples of size 2 *without replacement*.  This means a value can only appear once in each sample.\r<BR>\n" );
document.write( "\n" );
document.write( "* **Samples:** (12, 12), (12, 14), (12, 16), (14, 16)<BR>\n" );
document.write( "* **Sample Means:** 12, 13, 14, 15\r<BR>\n" );
document.write( "\n" );
document.write( "**B. Comparing the Mean of Sample Means and the Population Mean:**\r<BR>\n" );
document.write( "\n" );
document.write( "* **Population Mean:** (12 + 12 + 14 + 16) / 4 = 13.5<BR>\n" );
document.write( "* **Mean of Sample Means:** (12 + 13 + 14 + 15) / 4 = 13.5\r<BR>\n" );
document.write( "\n" );
document.write( "The mean of the distribution of sample means is equal to the population mean. This is a fundamental concept in statistics.\r<BR>\n" );
document.write( "\n" );
document.write( "**C. Comparing Dispersion:**\r<BR>\n" );
document.write( "\n" );
document.write( "* **Population Standard Deviation:**  We can calculate the population standard deviation using the following formula:<BR>\n" );
document.write( "   σ = sqrt[ Σ(xi - μ)² / N ]<BR>\n" );
document.write( "   Where:<BR>\n" );
document.write( "     * xi are the individual values<BR>\n" );
document.write( "     * μ is the population mean<BR>\n" );
document.write( "     * N is the population size\r<BR>\n" );
document.write( "\n" );
document.write( "   σ = sqrt[ (1.5² + 1.5² + 0.5² + 2.5²) / 4 ]<BR>\n" );
document.write( "   σ = sqrt[ 2.25+2.25+0.25+6.25 /4] = sqrt(11/4) = sqrt(2.75) ≈ 1.66\r<BR>\n" );
document.write( "\n" );
document.write( "* **Standard Deviation of Sample Means (Standard Error):** The standard deviation of the sample means is called the standard error. It is calculated as:<BR>\n" );
document.write( "   Standard Error = σ / sqrt(n)<BR>\n" );
document.write( "    Where σ is the population standard deviation and n is the sample size. \r<BR>\n" );
document.write( "\n" );
document.write( "   Standard Error = 1.66 / sqrt(2) ≈ 1.17\r<BR>\n" );
document.write( "\n" );
document.write( "The dispersion (as measured by the standard deviation) of the sample means is *smaller* than the dispersion of the population.  This is also a fundamental concept: the distribution of sample means is less variable than the original population. This makes sense  -  averages tend to be more stable and less extreme than individual values.<BR>\n" );
document.write( " </SPAN>\n" );
document.write( "</TD></TR></TABLE>\n" );