document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1176053>Question 1176053</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Suppose a random sample of size n = 10 is taken from a normally distributed population with variance sigma^2 = 1.5.\r<BR>\n" );
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document.write( "a. Use a Chi-Squared table to approximate the probability that the sample variance is less than 2.446.  That is, approximate P(S^2 <= 2.446) where S^2 is the sample variance.  (HINT: According to theorem the random variable C = (10-1)S^2/1.5 has a Chi-Squared(9) distribution.  This means that the event S^2 <= 2.446 corresponds to the event C = 9S^2/1.5 <= 9(2.446)/1.5 = 14.68.  Thus P(S^2 <= 2.446) = P(C <=14.68))\r<BR>\n" );
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document.write( "b. Now suppose the sample size is n = 31.  Approximate P(S^2 <= 0.83955) </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #850586 by </B><A HREF=/tutors/aboutme.mpl?userid=CPhill><B>CPhill(2189)</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=850586\">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( "**a. n = 10, S^2 <= 2.446**\r<BR>\n" );
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document.write( "1.  **Calculate the Chi-Squared Statistic (C):**<BR>\n" );
document.write( "    * C = (n - 1) * S^2 / σ^2<BR>\n" );
document.write( "    * C = (10 - 1) * 2.446 / 1.5<BR>\n" );
document.write( "    * C = 9 * 2.446 / 1.5<BR>\n" );
document.write( "    * C = 22.014 / 1.5<BR>\n" );
document.write( "    * C = 14.676 (which rounds to 14.68 as given in the problem)\r<BR>\n" );
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document.write( "2.  **Degrees of Freedom (df):**<BR>\n" );
document.write( "    * df = n - 1 = 10 - 1 = 9\r<BR>\n" );
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document.write( "3.  **Find the Probability Using the Chi-Squared Table:**<BR>\n" );
document.write( "    * We need to find P(C ≤ 14.68) with 9 degrees of freedom.<BR>\n" );
document.write( "    * Look up 14.68 in the Chi-Squared distribution table with 9 degrees of freedom.<BR>\n" );
document.write( "    * Using a Chi-Squared table, we find that 14.68 falls between the values corresponding to cumulative probabilities of 0.10 and 0.05.<BR>\n" );
document.write( "    * More specifically, for df = 9, the chi-square value for 0.10 is 14.684, and for 0.05 it is 16.919.<BR>\n" );
document.write( "    * Since 14.676 is very close to 14.684, the probability P(C ≤ 14.68) is approximately 0.10.\r<BR>\n" );
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document.write( "**Therefore, P(S^2 ≤ 2.446) ≈ 0.10**\r<BR>\n" );
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document.write( "**b. n = 31, S^2 <= 0.83955**\r<BR>\n" );
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document.write( "1.  **Calculate the Chi-Squared Statistic (C):**<BR>\n" );
document.write( "    * C = (n - 1) * S^2 / σ^2<BR>\n" );
document.write( "    * C = (31 - 1) * 0.83955 / 1.5<BR>\n" );
document.write( "    * C = 30 * 0.83955 / 1.5<BR>\n" );
document.write( "    * C = 25.1865 / 1.5<BR>\n" );
document.write( "    * C = 16.791\r<BR>\n" );
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document.write( "2.  **Degrees of Freedom (df):**<BR>\n" );
document.write( "    * df = n - 1 = 31 - 1 = 30\r<BR>\n" );
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document.write( "3.  **Find the Probability Using the Chi-Squared Table or Scipy:**<BR>\n" );
document.write( "    * We need to find P(C ≤ 16.791) with 30 degrees of freedom.\r<BR>\n" );
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document.write( "    * Using a Chi-Squared table, we find that the value 16.791 falls between the values corresponding to cumulative probabilities of 0.025 and 0.01.<BR>\n" );
document.write( "    * More specifically, for df = 30, the chi-square value for 0.025 is 16.791.\r<BR>\n" );
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document.write( "    * Alternatively, we can use Python's Scipy library, as illustrated in the provided code, which calculates the cumulative distribution function (CDF) of the Chi-Squared distribution.\r<BR>\n" );
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document.write( "    * Based on the provided code, P(S^2 <= 0.83955) = 0.0250\r<BR>\n" );
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document.write( "**Therefore, P(S^2 ≤ 0.83955) ≈ 0.025**<BR>\n" );
document.write( " </SPAN>\n" );
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