document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1178447>Question 1178447</A>: <I><SPAN STYLE=\"word-break: break-all;\"> A particular type of vacuum-packed coffee packet contains an average of 16 ounces. It has been observed that the number of ounces of coffee in these packets is normally distributed with σ = 1.41 ounce. A random sample of 15 of these coffee packets is selected, and the observations are used to calculate s. Find the numbers a and b such that P (a ≤ S^2 ≤ b) = 0.90. \r<BR>\n" );
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document.write( "thank you  </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #850345 by </B><A HREF=/tutors/aboutme.mpl?userid=CPhill><B>CPhill(1959)</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=850345\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> Let's break down how to find the values 'a' and 'b' such that P(a ≤ S² ≤ b) = 0.90, where S² is the sample variance.\r<BR>\n" );
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document.write( "**Understanding the Problem**\r<BR>\n" );
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document.write( "* **Population Distribution:** The ounces of coffee are normally distributed with a standard deviation (σ) of 1.41 ounces.<BR>\n" );
document.write( "* **Sample Size:** A random sample of 15 packets is taken (n = 15).<BR>\n" );
document.write( "* **Sample Variance (S²):** We're interested in the distribution of the sample variance.<BR>\n" );
document.write( "* **Chi-Square Distribution:** The quantity (n - 1)S² / σ² follows a chi-square distribution with (n - 1) degrees of freedom.<BR>\n" );
document.write( "* **Goal:** Find 'a' and 'b' such that the probability that S² falls between them is 0.90.\r<BR>\n" );
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document.write( "**Steps to Solve**\r<BR>\n" );
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document.write( "1.  **Degrees of Freedom:**<BR>\n" );
document.write( "    * The degrees of freedom (df) are n - 1 = 15 - 1 = 14.\r<BR>\n" );
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document.write( "2.  **Chi-Square Distribution:**<BR>\n" );
document.write( "    * We know that (14 * S²) / (1.41²) ~ χ²(14).\r<BR>\n" );
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document.write( "3.  **Finding Chi-Square Percentiles:**<BR>\n" );
document.write( "    * We want to find values χ²_lower and χ²_upper such that P(χ²_lower ≤ χ²(14) ≤ χ²_upper) = 0.90.<BR>\n" );
document.write( "    * Since we want the middle 90%, we need to exclude the lower 5% and the upper 5%.<BR>\n" );
document.write( "    * Therefore, we need to find the 5th and 95th percentiles of the chi-square distribution with 14 degrees of freedom.\r<BR>\n" );
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document.write( "4.  **Calculating 'a' and 'b':**<BR>\n" );
document.write( "    * Let χ²_0.05 be the 5th percentile and χ²_0.95 be the 95th percentile.<BR>\n" );
document.write( "    * We have:<BR>\n" );
document.write( "        * χ²_0.05 = (14 * b) / (1.41²)<BR>\n" );
document.write( "        * χ²_0.95 = (14 * a) / (1.41²)<BR>\n" );
document.write( "    * Therefore:<BR>\n" );
document.write( "        * a = (14 * 1.41²) / χ²_0.95<BR>\n" );
document.write( "        * b = (14 * 1.41²) / χ²_0.05\r<BR>\n" );
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document.write( "5.  **Using scipy.stats:**<BR>\n" );
document.write( "    * Using python and the scipy.stats library, we can compute the answer.\r<BR>\n" );
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document.write( "```python<BR>\n" );
document.write( "import scipy.stats as stats\r<BR>\n" );
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document.write( "# Sample size<BR>\n" );
document.write( "n = 15\r<BR>\n" );
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document.write( "# Population standard deviation<BR>\n" );
document.write( "sigma = 1.41\r<BR>\n" );
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document.write( "# Degrees of freedom<BR>\n" );
document.write( "df = n - 1 \r<BR>\n" );
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document.write( "# Find the chi-square values for the 5th and 95th percentiles<BR>\n" );
document.write( "a = (df * sigma**2) / stats.chi2.ppf(0.95, df)<BR>\n" );
document.write( "b = (df * sigma**2) / stats.chi2.ppf(0.05, df)\r<BR>\n" );
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document.write( "print(f\"a: {a:.4f}\")<BR>\n" );
document.write( "print(f\"b: {b:.4f}\")<BR>\n" );
document.write( "```\r<BR>\n" );
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document.write( "6.  **Results**<BR>\n" );
document.write( "    * a ≈ 1.1752<BR>\n" );
document.write( "    * b ≈ 4.2360\r<BR>\n" );
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document.write( "**Conclusion**\r<BR>\n" );
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document.write( "The values of 'a' and 'b' are approximately 1.1752 and 4.2360, respectively. This means that there is a 90% probability that the sample variance (S²) will fall between these two values.<BR>\n" );
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