document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1166922>Question 1166922</A>: <I><SPAN STYLE=\"word-break: break-all;\"> The distribution of scores on a standardized aptitude test is approximately normal with a mean of 520 and a standard deviation of 95. \r<BR>\n" );
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document.write( "What is the minimum score needed to be in the top 15% on this test? Carry your intermediate computations to at least four decimal places, and round your answer to the nearest integer. </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #852690 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=852690\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> ```python?code_reference&code_event_index=2<BR>\n" );
document.write( "import numpy as np<BR>\n" );
document.write( "from scipy.stats import norm\r<BR>\n" );
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document.write( "# Given parameters<BR>\n" );
document.write( "mu = 520<BR>\n" );
document.write( "sigma = 95<BR>\n" );
document.write( "percentile = 0.85 # Corresponds to the top 15% (100% - 15% = 85%)\r<BR>\n" );
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document.write( "# 1. Find the z-score corresponding to the 85th percentile<BR>\n" );
document.write( "# Use the Percent Point Function (inverse of CDF)<BR>\n" );
document.write( "z_score = norm.ppf(percentile)\r<BR>\n" );
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document.write( "# 2. Calculate the minimum score (X)<BR>\n" );
document.write( "# X = mu + Z * sigma<BR>\n" );
document.write( "X = mu + z_score * sigma\r<BR>\n" );
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document.write( "# 3. Print the intermediate z-score and the final score X<BR>\n" );
document.write( "print(f\"Mean (mu): {mu}\")<BR>\n" );
document.write( "print(f\"Standard Deviation (sigma): {sigma}\")<BR>\n" );
document.write( "print(f\"Cumulative Probability: {percentile}\")<BR>\n" );
document.write( "print(f\"Z-score (intermediate): {z_score:.4f}\")<BR>\n" );
document.write( "print(f\"Raw Score (X): {X}\")\r<BR>\n" );
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document.write( "# Round the final answer to the nearest integer<BR>\n" );
document.write( "final_score = round(X)<BR>\n" );
document.write( "print(f\"Final score (rounded to nearest integer): {final_score}\")\r<BR>\n" );
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document.write( "```\r<BR>\n" );
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document.write( "```text?code_stdout&code_event_index=2<BR>\n" );
document.write( "Mean (mu): 520<BR>\n" );
document.write( "Standard Deviation (sigma): 95<BR>\n" );
document.write( "Cumulative Probability: 0.85<BR>\n" );
document.write( "Z-score (intermediate): 1.0364<BR>\n" );
document.write( "Raw Score (X): 618.46117200191<BR>\n" );
document.write( "Final score (rounded to nearest integer): 618\r<BR>\n" );
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document.write( "```\r<BR>\n" );
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document.write( "The minimum score needed to be in the top $15\%$ on this test is **618**.\r<BR>\n" );
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document.write( "This is calculated by finding the score that corresponds to the **$85^{th}$ percentile** of the normal distribution, since the top $15\%$ of scores are those above $100\% - 15\% = 85\%$.\r<BR>\n" );
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document.write( "-----\r<BR>\n" );
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document.write( "## Calculation Steps\r<BR>\n" );
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document.write( "### 1\. Find the Z-score\r<BR>\n" );
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document.write( "The first step is to find the **Z-score** ($Z$) that has $85\%$ of the area under the standard normal curve to its left.\r<BR>\n" );
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document.write( "Using a Z-table or statistical software (carrying intermediate computations to at least four decimal places):<BR>\n" );
document.write( "$$Z \approx \mathbf{1.0364}$$\r<BR>\n" );
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document.write( "### 2\. Calculate the Raw Score\r<BR>\n" );
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document.write( "We use the Z-score formula rearranged to solve for the raw score ($X$):\r<BR>\n" );
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document.write( "$$X = \mu + Z \cdot \sigma$$\r<BR>\n" );
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document.write( "Where:\r<BR>\n" );
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document.write( "  * $\mu$ (Mean) $= 520$<BR>\n" );
document.write( "  * $\sigma$ (Standard Deviation) $= 95$<BR>\n" );
document.write( "  * $Z$ (Z-score) $\approx 1.0364$\r<BR>\n" );
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document.write( "Substituting the values:<BR>\n" );
document.write( "$$X = 520 + (1.0364) \cdot 95$$<BR>\n" );
document.write( "$$X = 520 + 98.458$$<BR>\n" );
document.write( "$$X \approx 618.458$$\r<BR>\n" );
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document.write( "### 3\. Round to the Nearest Integer\r<BR>\n" );
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document.write( "Rounding the result to the nearest integer:<BR>\n" );
document.write( "$$618.458 \approx \mathbf{618}$$\r<BR>\n" );
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document.write( "A score of **618** is the minimum score required to be in the top $15\%$ of test takers. </SPAN>\n" );
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