document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1164810>Question 1164810</A>: <I><SPAN STYLE=\"word-break: break-all;\"> In grading pears into small, medium, and large, Nature Farms packs the pears that weigh more than 3.6 ounces in packages marked \"large\" and the pears that weigh less than 2.4 ounces into packages marked \"small\"; the remainder are packed in packages marked \"medium.\" If a day's packaging contained 3.92% large and 5.94% small pears, determine the mean and the standard deviation for the pears' weights. Assume that the distribution of the weights is normal </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #853515 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=853515\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> ```python<BR>\n" );
document.write( "import scipy.stats as stats\r<BR>\n" );
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document.write( "# Probabilities<BR>\n" );
document.write( "p_large = 0.0392<BR>\n" );
document.write( "p_small = 0.0594\r<BR>\n" );
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document.write( "# Cumulative probabilities for Z<BR>\n" );
document.write( "cp_large = 1 - p_large  # P(X < 3.6)<BR>\n" );
document.write( "cp_small = p_small      # P(X < 2.4)\r<BR>\n" );
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document.write( "# Z-scores<BR>\n" );
document.write( "z1 = stats.norm.ppf(cp_large) # For 3.6<BR>\n" );
document.write( "z2 = stats.norm.ppf(cp_small) # For 2.4\r<BR>\n" );
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document.write( "# System of equations:<BR>\n" );
document.write( "# 3.6 = mu + z1 * sigma<BR>\n" );
document.write( "# 2.4 = mu + z2 * sigma<BR>\n" );
document.write( "# Subtracting: 1.2 = (z1 - z2) * sigma\r<BR>\n" );
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document.write( "sigma = 1.2 / (z1 - z2)<BR>\n" );
document.write( "mu = 3.6 - z1 * sigma\r<BR>\n" );
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document.write( "print(f\"{z1=}\")<BR>\n" );
document.write( "print(f\"{z2=}\")<BR>\n" );
document.write( "print(f\"{sigma=}\")<BR>\n" );
document.write( "print(f\"{mu=}\")\r<BR>\n" );
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document.write( "```\r<BR>\n" );
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document.write( "```text<BR>\n" );
document.write( "z1=1.7600460440337644<BR>\n" );
document.write( "z2=-1.559830253819668<BR>\n" );
document.write( "sigma=0.36145925098953136<BR>\n" );
document.write( "mu=2.963815075216468\r<BR>\n" );
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document.write( "```\r<BR>\n" );
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document.write( "To find the mean () and standard deviation () of the pears' weights, we use the properties of the normal distribution and set up a system of linear equations based on the given probabilities.\r<BR>\n" );
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document.write( "### 1. Identify the given information\r<BR>\n" );
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document.write( "* **Large pears:** <BR>\n" );
document.write( "* **Small pears:** \r<BR>\n" );
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document.write( "### 2. Find the corresponding -scores\r<BR>\n" );
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document.write( "We use the standard normal distribution table (or inverse cumulative distribution function) to find the -scores that correspond to these probabilities.\r<BR>\n" );
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document.write( "* For **Large pears**:\r<BR>\n" );
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document.write( "The -score such that  is ****.<BR>\n" );
document.write( "* For **Small pears**:\r<BR>\n" );
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document.write( "The -score such that  is ****.\r<BR>\n" );
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document.write( "### 3. Set up the equations\r<BR>\n" );
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document.write( "Using the formula :\r<BR>\n" );
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document.write( "1. <BR>\n" );
document.write( "2. \r<BR>\n" );
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document.write( "### 4. Solve the system of equations\r<BR>\n" );
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document.write( "Subtract the second equation from the first to eliminate :\r<BR>\n" );
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document.write( "Now, substitute  back into the first equation to find :\r<BR>\n" );
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document.write( "### Final Answer:\r<BR>\n" );
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document.write( "* **Mean ():**  ounces<BR>\n" );
document.write( "* **Standard Deviation ():**  ounces\r<BR>\n" );
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document.write( "The distribution of the pears' weights is ****. </SPAN>\n" );
document.write( "</TD></TR></TABLE>\n" );