document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1167512>Question 1167512</A>: <I><SPAN STYLE=\"word-break: break-all;\"> The results of the latest writing of the LSAT (Law School Aptitude Test) showed results that were normally distributed with a mean score of 826 and a standard deviation of 50.\r<BR>\n" );
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document.write( "For part (d) enter probability answers in decimal form (i.e. 0.0003 instead of 0.0300  %<BR>\n" );
document.write( " ). Round the final answer to 4 decimals.\r<BR>\n" );
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document.write( " c)If a group of 59 applicants is randomly selected, what is the probability that the group average is not less than 846? </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #852326 by </B><A HREF=/tutors/aboutme.mpl?userid=CPhill><B>CPhill(1987)</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=852326\">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( "# Define parameters<BR>\n" );
document.write( "mu = 826    # Population mean score<BR>\n" );
document.write( "sigma = 50  # Population standard deviation<BR>\n" );
document.write( "n = 59      # Sample size<BR>\n" );
document.write( "X_bar = 846 # Sample average score\r<BR>\n" );
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document.write( "# 1. Calculate the Standard Error (Standard Deviation of the Sample Mean)<BR>\n" );
document.write( "sigma_x_bar = sigma / np.sqrt(n)\r<BR>\n" );
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document.write( "# 2. Calculate the Z-score for X_bar = 846<BR>\n" );
document.write( "Z = (X_bar - mu) / sigma_x_bar\r<BR>\n" );
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document.write( "# 3. Calculate the probability P(X_bar >= 846)<BR>\n" );
document.write( "# P(X_bar >= 846) = 1 - P(X_bar < 846) = 1 - norm.cdf(Z)<BR>\n" );
document.write( "probability = 1 - norm.cdf(Z)\r<BR>\n" );
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document.write( "# Print results rounded to 4 decimal places<BR>\n" );
document.write( "print(f\"Population Mean (mu): {mu}\")<BR>\n" );
document.write( "print(f\"Population Standard Deviation (sigma): {sigma}\")<BR>\n" );
document.write( "print(f\"Sample Size (n): {n}\")<BR>\n" );
document.write( "print(f\"Standard Error (sigma_x_bar): {sigma_x_bar:.4f}\")<BR>\n" );
document.write( "print(f\"Z-score: {Z:.4f}\")<BR>\n" );
document.write( "print(f\"Probability P(X_bar >= 846): {probability:.4f}\")\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( "Population Mean (mu): 826<BR>\n" );
document.write( "Population Standard Deviation (sigma): 50<BR>\n" );
document.write( "Sample Size (n): 59<BR>\n" );
document.write( "Standard Error (sigma_x_bar): 6.5094<BR>\n" );
document.write( "Z-score: 3.0725<BR>\n" );
document.write( "Probability P(X_bar >= 846): 0.0011\r<BR>\n" );
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document.write( "```\r<BR>\n" );
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document.write( "The probability that the group average score is not less than $846$ is **0.0011**.\r<BR>\n" );
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document.write( "Since the individual scores are normally distributed, the **Central Limit Theorem** tells us that the distribution of the sample mean ($\bar{X}$) is also normally distributed.\r<BR>\n" );
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document.write( "Here is the step-by-step calculation:\r<BR>\n" );
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document.write( "### 1\. Identify the Distribution of the Sample Mean\r<BR>\n" );
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document.write( "  * Population Mean ($\mu$): $826$<BR>\n" );
document.write( "  * Population Standard Deviation ($\sigma$): $50$<BR>\n" );
document.write( "  * Sample Size ($n$): $59$<BR>\n" );
document.write( "  * Target Sample Mean ($\bar{X}$): $846$\r<BR>\n" );
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document.write( "The mean of the sample averages is:<BR>\n" );
document.write( "$$\mu_{\bar{X}} = \mu = 826$$\r<BR>\n" );
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document.write( "The standard deviation of the sample averages (Standard Error) is:<BR>\n" );
document.write( "$$\sigma_{\bar{X}} = \frac{\sigma}{\sqrt{n}} = \frac{50}{\sqrt{59}} \approx 6.5094$$\r<BR>\n" );
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document.write( "### 2\. Calculate the Z-score\r<BR>\n" );
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document.write( "We standardize the target average ($\bar{X} = 846$):<BR>\n" );
document.write( "$$Z = \frac{\bar{X} - \mu_{\bar{X}}}{\sigma_{\bar{X}}} = \frac{846 - 826}{50/\sqrt{59}} \approx \frac{20}{6.5094} \approx 3.0725$$\r<BR>\n" );
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document.write( "### 3\. Calculate the Probability\r<BR>\n" );
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document.write( "The question asks for the probability that the group average is **not less than** $846$, which means $P(\bar{X} \ge 846)$ or $P(Z \ge 3.0725)$.\r<BR>\n" );
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document.write( "$$P(\bar{X} \ge 846) = 1 - P(\bar{X} < 846) = 1 - \Phi(3.0725)$$\r<BR>\n" );
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document.write( "Using the Z-score value:<BR>\n" );
document.write( "$$1 - 0.998910 \approx 0.001090$$\r<BR>\n" );
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document.write( "Rounding the final answer to $4$ decimal places: **0.0011**. </SPAN>\n" );
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