document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1190564>Question 1190564</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Hello, Good evening, I need help on my homework. I am having a hard time doing Random Sampling\r<BR>\n" );
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document.write( "A population consists of (3,8,10,15). Consider all possible sample size 2 that can be drawn with replacement.\r<BR>\n" );
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document.write( "-what is the standard deviation of the sampling distribution of means? = \r<BR>\n" );
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document.write( "-what is the population mean? =\r<BR>\n" );
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document.write( "-what is the population standard deviation? = \r<BR>\n" );
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document.write( "-what is the population variance? = \r<BR>\n" );
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document.write( "-what is the variance of the sampling distribution of means? =\r<BR>\n" );
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document.write( "-what is the mean of the sampling distribution of means? =\r<BR>\n" );
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document.write( "A population consists of (3,8,10,15). Consider all possible sample size<BR>\n" );
document.write( " 2 that can be drawn without replacement.\r<BR>\n" );
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document.write( "-what is the variance of the sampling distribution of means? =\r<BR>\n" );
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document.write( "-what is the standard deviation of the sampling distribution of means? = \r<BR>\n" );
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document.write( "-what is the mean of the sampling distribution of means? = \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 #822328 by </B><A HREF=/tutors/aboutme.mpl?userid=chitrank><B>chitrank(4)</B></A><img src=\"/images/stars/iconNewId_16x16.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=chitrank><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/images/nopicture.jpg><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=822328\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> Let the first sample be denoted by random variable <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=X_1 ALIGN=MIDDLE  ALT=\"X_1\"   DISABLED_style= \"max-width:90%; height:auto;\" > and the second sample by <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=X_2 ALIGN=MIDDLE  ALT=\"X_2\"   DISABLED_style= \"max-width:90%; height:auto;\" >. <BR>\n" );
document.write( "First few notes about <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=X_1 ALIGN=MIDDLE  ALT=\"X_1\"   DISABLED_style= \"max-width:90%; height:auto;\" >and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=X_2 ALIGN=MIDDLE  ALT=\"X_2\"   DISABLED_style= \"max-width:90%; height:auto;\" >. <BR>\n" );
document.write( "---1 They are indepdent RVs. Why? Because sampling happens with replacement. Which means, if I sample <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=X_1 ALIGN=MIDDLE  ALT=\"X_1\"   DISABLED_style= \"max-width:90%; height:auto;\" >, then no matter what<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=X_1 ALIGN=MIDDLE  ALT=\"X_1\"   DISABLED_style= \"max-width:90%; height:auto;\" > really was, the distribution of <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=X_2 ALIGN=MIDDLE  ALT=\"X_2\"   DISABLED_style= \"max-width:90%; height:auto;\" > remains the same.\r<BR>\n" );
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document.write( "Now consider the random variable <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=X_3=%28X_1%2BX_2%29%2F2 ALIGN=MIDDLE  ALT=\"X_3=%28X_1%2BX_2%29%2F2\"   DISABLED_style= \"max-width:90%; height:auto;\" >. Do you see why <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=X_3 ALIGN=MIDDLE  ALT=\"X_3\"   DISABLED_style= \"max-width:90%; height:auto;\" > is the sample mean?\r<BR>\n" );
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document.write( "Now we can simply apply formulas and theorems to calculate each of the quantities that you have asked.<BR>\n" );
document.write( "First, what is mean of the sample-mean? Easy. As you may know, \"mean\" of a random variable is nothing but an \"expectation\" operator \"E\" on the random variable. Therefore mean of the sample means would be <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=E%28X_3%29 ALIGN=MIDDLE  ALT=\"E%28X_3%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > which is <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=E%28%28X_1%2BX_2%29%2F2%29 ALIGN=MIDDLE  ALT=\"E%28%28X_1%2BX_2%29%2F2%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >. Remember <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=E ALIGN=MIDDLE  ALT=\"E\"   DISABLED_style= \"max-width:90%; height:auto;\" > is a linear operator? Which gives us <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=E%28X_3%29=%28E%28X_1%29%2BE%28X_2%29%29%2F2 ALIGN=MIDDLE  ALT=\"E%28X_3%29=%28E%28X_1%29%2BE%28X_2%29%29%2F2\"   DISABLED_style= \"max-width:90%; height:auto;\" >. What is <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=E%28X_1%29 ALIGN=MIDDLE  ALT=\"E%28X_1%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >, or mean of the population? It is <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%283%2B8%2B10%2B15%29%2F4 ALIGN=MIDDLE  ALT=\"%283%2B8%2B10%2B15%29%2F4\"   DISABLED_style= \"max-width:90%; height:auto;\" > which is <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=36%2F4=9 ALIGN=MIDDLE  ALT=\"36%2F4=9\"   DISABLED_style= \"max-width:90%; height:auto;\" >. What is <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=E%28X_2%29 ALIGN=MIDDLE  ALT=\"E%28X_2%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >? Well that is also 9 (Remember we are sampling with replacement?)  </SPAN>\n" );
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