document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1174175>Question 1174175</A>: <I><SPAN STYLE=\"word-break: break-all;\"> TASK 1: Solve for the mean μ, variance σ^2 of the population and the mean μ_¯x and variance 〖σ^2〗_¯x of the sampling distribution of the sample means.\r<BR>\n" );
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document.write( "1. A population consists of the five measurements: 2, 6, 8, 0 and 3. Suppose samples of size 2 are drawn from this population. </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #850706 by </B><A HREF=/tutors/aboutme.mpl?userid=CPhill><B>CPhill(1987)</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=850706\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> Let's solve this problem step-by-step.\r<BR>\n" );
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document.write( "**1. Calculate the Population Mean (μ)**\r<BR>\n" );
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document.write( "* Population: 2, 6, 8, 0, 3<BR>\n" );
document.write( "* Sum of the population: 2 + 6 + 8 + 0 + 3 = 19<BR>\n" );
document.write( "* Number of elements in the population: 5<BR>\n" );
document.write( "* Population mean (μ) = Sum / Number of elements = 19 / 5 = 3.8\r<BR>\n" );
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document.write( "**2. Calculate the Population Variance (σ^2)**\r<BR>\n" );
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document.write( "* First, find the squared differences from the mean:<BR>\n" );
document.write( "    * (2 - 3.8)^2 = (-1.8)^2 = 3.24<BR>\n" );
document.write( "    * (6 - 3.8)^2 = (2.2)^2 = 4.84<BR>\n" );
document.write( "    * (8 - 3.8)^2 = (4.2)^2 = 17.64<BR>\n" );
document.write( "    * (0 - 3.8)^2 = (-3.8)^2 = 14.44<BR>\n" );
document.write( "    * (3 - 3.8)^2 = (-0.8)^2 = 0.64<BR>\n" );
document.write( "* Sum of squared differences: 3.24 + 4.84 + 17.64 + 14.44 + 0.64 = 40.8<BR>\n" );
document.write( "* Population variance (σ^2) = Sum of squared differences / Number of elements = 40.8 / 5 = 8.16\r<BR>\n" );
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document.write( "**3. Generate All Possible Samples of Size 2**\r<BR>\n" );
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document.write( "* We need to find all combinations of 2 elements from the population of 5.<BR>\n" );
document.write( "* These samples are:<BR>\n" );
document.write( "    * (2, 6)<BR>\n" );
document.write( "    * (2, 8)<BR>\n" );
document.write( "    * (2, 0)<BR>\n" );
document.write( "    * (2, 3)<BR>\n" );
document.write( "    * (6, 8)<BR>\n" );
document.write( "    * (6, 0)<BR>\n" );
document.write( "    * (6, 3)<BR>\n" );
document.write( "    * (8, 0)<BR>\n" );
document.write( "    * (8, 3)<BR>\n" );
document.write( "    * (0, 3)\r<BR>\n" );
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document.write( "**4. Calculate the Mean of Each Sample (Sample Means)**\r<BR>\n" );
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document.write( "* Calculate the mean of each of the samples.<BR>\n" );
document.write( "    * (2, 6) mean = 4<BR>\n" );
document.write( "    * (2, 8) mean = 5<BR>\n" );
document.write( "    * (2, 0) mean = 1<BR>\n" );
document.write( "    * (2, 3) mean = 2.5<BR>\n" );
document.write( "    * (6, 8) mean = 7<BR>\n" );
document.write( "    * (6, 0) mean = 3<BR>\n" );
document.write( "    * (6, 3) mean = 4.5<BR>\n" );
document.write( "    * (8, 0) mean = 4<BR>\n" );
document.write( "    * (8, 3) mean = 5.5<BR>\n" );
document.write( "    * (0, 3) mean = 1.5\r<BR>\n" );
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document.write( "**5. Calculate the Mean of the Sample Means (μ_¯x)**\r<BR>\n" );
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document.write( "* Sum of sample means: 4 + 5 + 1 + 2.5 + 7 + 3 + 4.5 + 4 + 5.5 + 1.5 = 38<BR>\n" );
document.write( "* Number of samples: 10<BR>\n" );
document.write( "* Mean of sample means (μ_¯x) = 38 / 10 = 3.8\r<BR>\n" );
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document.write( "**6. Calculate the Variance of the Sample Means (σ^2_¯x)**\r<BR>\n" );
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document.write( "* Find the squared differences from μ_¯x (3.8):<BR>\n" );
document.write( "    * (4 - 3.8)^2 = 0.04<BR>\n" );
document.write( "    * (5 - 3.8)^2 = 1.44<BR>\n" );
document.write( "    * (1 - 3.8)^2 = 7.84<BR>\n" );
document.write( "    * (2.5 - 3.8)^2 = 1.69<BR>\n" );
document.write( "    * (7 - 3.8)^2 = 10.24<BR>\n" );
document.write( "    * (3 - 3.8)^2 = 0.64<BR>\n" );
document.write( "    * (4.5 - 3.8)^2 = 0.49<BR>\n" );
document.write( "    * (4 - 3.8)^2 = 0.04<BR>\n" );
document.write( "    * (5.5 - 3.8)^2 = 2.89<BR>\n" );
document.write( "    * (1.5 - 3.8)^2 = 5.29<BR>\n" );
document.write( "* Sum of squared differences: 0.04 + 1.44 + 7.84 + 1.69 + 10.24 + 0.64 + 0.49 + 0.04 + 2.89 + 5.29 = 30.6<BR>\n" );
document.write( "* Variance of sample means (σ^2_¯x) = 30.6 / 10 = 3.06\r<BR>\n" );
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document.write( "**Results:**\r<BR>\n" );
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document.write( "* Population Mean (μ): 3.8<BR>\n" );
document.write( "* Population Variance (σ^2): 8.16<BR>\n" );
document.write( "* Mean of Sample Means (μ_¯x): 3.8<BR>\n" );
document.write( "* Variance of Sample Means (σ^2_¯x): 3.06<BR>\n" );
document.write( " </SPAN>\n" );
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