document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1171230>Question 1171230</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Suppose (X1,X2,…,X9) is a random sample from Normal(2,42). Let X(bar) be the sample mean of X1,X2,…,X9 and S2 be the sample variance of X1,X2,…,X9. For items asking for the distribution of a statistic, do not forget to specify the parameters.\r<BR>\n" );
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document.write( "What is the distribution of the sample mean (X(bar))? </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #850975 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=850975\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> Given that $X_1, X_2, \ldots, X_9$ is a random sample from a normal distribution with mean $\mu = 2$ and variance $\sigma^2 = 42$, i.e., $X_i \sim N(2, 42)$ for $i=1, 2, \ldots, 9$, we want to find the distribution of the sample mean $\bar{X}$.\r<BR>\n" );
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document.write( "The sample mean $\bar{X}$ is given by:\r<BR>\n" );
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document.write( "$$ \bar{X} = \frac{1}{n} \sum_{i=1}^{n} X_i $$\r<BR>\n" );
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document.write( "where $n = 9$ in this case.\r<BR>\n" );
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document.write( "Since the $X_i$ are normally distributed, the sample mean $\bar{X}$ is also normally distributed.\r<BR>\n" );
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document.write( "The mean of $\bar{X}$ is:\r<BR>\n" );
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document.write( "$$ E(\bar{X}) = E\left(\frac{1}{n} \sum_{i=1}^{n} X_i\right) = \frac{1}{n} \sum_{i=1}^{n} E(X_i) = \frac{1}{n} (n\mu) = \mu $$\r<BR>\n" );
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document.write( "In this case, $E(\bar{X}) = 2$.\r<BR>\n" );
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document.write( "The variance of $\bar{X}$ is:\r<BR>\n" );
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document.write( "$$ Var(\bar{X}) = Var\left(\frac{1}{n} \sum_{i=1}^{n} X_i\right) = \frac{1}{n^2} \sum_{i=1}^{n} Var(X_i) = \frac{1}{n^2} (n\sigma^2) = \frac{\sigma^2}{n} $$\r<BR>\n" );
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document.write( "In this case, $Var(\bar{X}) = \frac{42}{9} = \frac{14}{3} \approx 4.67$.\r<BR>\n" );
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document.write( "Therefore, the distribution of the sample mean $\bar{X}$ is:\r<BR>\n" );
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document.write( "$$ \bar{X} \sim N\left(2, \frac{42}{9}\right) \quad \text{or} \quad \bar{X} \sim N\left(2, \frac{14}{3}\right) \quad \text{or} \quad \bar{X} \sim N(2, 4.67) $$\r<BR>\n" );
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document.write( "Final Answer: The final answer is $\boxed{N(2, 42/9)}$<BR>\n" );
document.write( " </SPAN>\n" );
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