document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1166864>Question 1166864</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Assume that the Weight of the students in your sample enrolled at NSU have an<BR>\n" );
document.write( "approximately Normal distribution. Find the probability that the mean Weight of your<BR>\n" );
document.write( "sample selected from the university is (Quantitative variable)<BR>\n" );
document.write( "A. Within 2 standard deviations from the mean.<BR>\n" );
document.write( "B. More than 1 standard deviation from the mean. <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 #852691 by </B><A HREF=/tutors/aboutme.mpl?userid=CPhill><B>CPhill(2030)</B></A><img src=\"/images/stars/stars-13.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=852691\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> The probability that the mean weight of your sample falls within a certain number of standard deviations from the population mean can be determined using the **Empirical Rule** (or 68-95-99.7 Rule) for a Normal distribution.\r<BR>\n" );
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document.write( "The problem implies that the distribution of the **sample mean** ($\bar{X}$) is also Normal (due to the original distribution being Normal), and we are measuring distances in terms of the standard deviation of the sample mean (the **Standard Error**, $\sigma_{\bar{X}}$).\r<BR>\n" );
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document.write( "## A. Within 2 Standard Deviations from the Mean\r<BR>\n" );
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document.write( "The probability that the sample mean is within **2 standard deviations** of the population mean ($\mu \pm 2\sigma_{\bar{X}}$) is approximately **95%**.\r<BR>\n" );
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document.write( "* $P(\mu - 2\sigma_{\bar{X}} < \bar{X} < \mu + 2\sigma_{\bar{X}}) \approx \mathbf{0.95}$ or $\mathbf{95\%}$\r<BR>\n" );
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document.write( "This value comes directly from the Empirical Rule, which states that approximately 95% of the data in a Normal distribution lies within two standard deviations of the mean. \r<BR>\n" );
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document.write( "[Image of a normal distribution curve with the area within 2 standard deviations shaded]\r<BR>\n" );
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document.write( "## B. More than 1 Standard Deviation from the Mean\r<BR>\n" );
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document.write( "The probability that the sample mean is **more than 1 standard deviation** from the mean is the area in the two tails combined (i.e., outside the range $\mu \pm 1\sigma_{\bar{X}}$).\r<BR>\n" );
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document.write( "1.  **Find the area within 1 standard deviation:**<BR>\n" );
document.write( "    According to the Empirical Rule, the probability that the sample mean is **within 1 standard deviation** of the population mean is approximately **68%**.<BR>\n" );
document.write( "    $$P(\mu - 1\sigma_{\bar{X}} < \bar{X} < \mu + 1\sigma_{\bar{X}}) \approx 0.68$$\r<BR>\n" );
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document.write( "2.  **Calculate the area outside 1 standard deviation:**<BR>\n" );
document.write( "    Since the total area under the probability curve is $1$ (or $100\%$), the probability of the sample mean being *more than* $1$ standard deviation away is $1$ minus the probability of it being *within* $1$ standard deviation.\r<BR>\n" );
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document.write( "    $$\text{Probability (More than 1 SD away)} = 1 - P(\text{Within 1 SD})$$<BR>\n" );
document.write( "    $$\text{Probability (More than 1 SD away)} \approx 1 - 0.68$$<BR>\n" );
document.write( "    $$\text{Probability (More than 1 SD away)} \approx \mathbf{0.32}$$ or $\mathbf{32\%}$\r<BR>\n" );
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document.write( "* $P(|\bar{X} - \mu| > 1\sigma_{\bar{X}}) \approx \mathbf{0.32}$ or $\mathbf{32\%}$ </SPAN>\n" );
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