document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1174088>Question 1174088</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Take CGPA of 05 students of your class; with n=2: <BR>\n" );
document.write( "Req: (a) Develop the sampling distribution of N and n<BR>\n" );
document.write( "	(b) Prove that mean of the means of samples is equal to mean of population<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 #850709 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=850709\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> Alright, let's work through this problem using hypothetical CGPA data for 5 students.\r<BR>\n" );
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document.write( "**1. Hypothetical CGPA Data**\r<BR>\n" );
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document.write( "Let's assume the CGPA of 5 students are:\r<BR>\n" );
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document.write( "* Student 1: 3.2<BR>\n" );
document.write( "* Student 2: 3.5<BR>\n" );
document.write( "* Student 3: 3.8<BR>\n" );
document.write( "* Student 4: 3.0<BR>\n" );
document.write( "* Student 5: 3.7\r<BR>\n" );
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document.write( "**2. Population Calculations**\r<BR>\n" );
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document.write( "* Population (N): {3.2, 3.5, 3.8, 3.0, 3.7}<BR>\n" );
document.write( "* Population Mean (μ): (3.2 + 3.5 + 3.8 + 3.0 + 3.7) / 5 = 17.2 / 5 = 3.44\r<BR>\n" );
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document.write( "**3. Sampling Distribution (n=2)**\r<BR>\n" );
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document.write( "* We need to find all possible samples of size 2 (n=2) from this population.<BR>\n" );
document.write( "* The samples are:<BR>\n" );
document.write( "    * (3.2, 3.5)<BR>\n" );
document.write( "    * (3.2, 3.8)<BR>\n" );
document.write( "    * (3.2, 3.0)<BR>\n" );
document.write( "    * (3.2, 3.7)<BR>\n" );
document.write( "    * (3.5, 3.8)<BR>\n" );
document.write( "    * (3.5, 3.0)<BR>\n" );
document.write( "    * (3.5, 3.7)<BR>\n" );
document.write( "    * (3.8, 3.0)<BR>\n" );
document.write( "    * (3.8, 3.7)<BR>\n" );
document.write( "    * (3.0, 3.7)\r<BR>\n" );
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document.write( "**4. Calculate Sample Means**\r<BR>\n" );
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document.write( "* Now, calculate the mean of each sample:<BR>\n" );
document.write( "    * (3.2 + 3.5) / 2 = 3.35<BR>\n" );
document.write( "    * (3.2 + 3.8) / 2 = 3.5<BR>\n" );
document.write( "    * (3.2 + 3.0) / 2 = 3.1<BR>\n" );
document.write( "    * (3.2 + 3.7) / 2 = 3.45<BR>\n" );
document.write( "    * (3.5 + 3.8) / 2 = 3.65<BR>\n" );
document.write( "    * (3.5 + 3.0) / 2 = 3.25<BR>\n" );
document.write( "    * (3.5 + 3.7) / 2 = 3.6<BR>\n" );
document.write( "    * (3.8 + 3.0) / 2 = 3.4<BR>\n" );
document.write( "    * (3.8 + 3.7) / 2 = 3.75<BR>\n" );
document.write( "    * (3.0 + 3.7) / 2 = 3.35\r<BR>\n" );
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document.write( "**5. Sampling Distribution of the Means**\r<BR>\n" );
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document.write( "* The sampling distribution of the means is:<BR>\n" );
document.write( "    * {3.35, 3.5, 3.1, 3.45, 3.65, 3.25, 3.6, 3.4, 3.75, 3.35}\r<BR>\n" );
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document.write( "**(a) Develop the sampling distribution of N and n**\r<BR>\n" );
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document.write( "* N = {3.2, 3.5, 3.8, 3.0, 3.7}<BR>\n" );
document.write( "* Sampling Distribution of n = {3.35, 3.5, 3.1, 3.45, 3.65, 3.25, 3.6, 3.4, 3.75, 3.35}\r<BR>\n" );
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document.write( "**(b) Prove that mean of the means of samples is equal to mean of population**\r<BR>\n" );
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document.write( "* Mean of the sample means (μ_x̄):<BR>\n" );
document.write( "    * (3.35 + 3.5 + 3.1 + 3.45 + 3.65 + 3.25 + 3.6 + 3.4 + 3.75 + 3.35) / 10 = 34.4 / 10 = 3.44<BR>\n" );
document.write( "* Population Mean (μ): 3.44\r<BR>\n" );
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document.write( "* μ_x̄ = μ = 3.44\r<BR>\n" );
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document.write( "**Therefore, the mean of the sample means (3.44) is equal to the population mean (3.44).**<BR>\n" );
document.write( " </SPAN>\n" );
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