document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1181470>Question 1181470</A>: <I><SPAN STYLE=\"word-break: break-all;\"> A distribution of values is normal with a mean of 199.9 and a standard deviation of 82.\r<BR>\n" );
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document.write( "Find P57, which is the score separating the bottom 57% from the top 43%.\r<BR>\n" );
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document.write( "P57 = ______\r<BR>\n" );
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document.write( "Enter your answer as a number accurate to 4 decimal places.  </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #850008 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=850008\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> Here's how to find P57:\r<BR>\n" );
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document.write( "1. **Find the z-score:** P57 represents the 57th percentile.  We need to find the z-score that corresponds to a cumulative probability of 0.57. You can use a z-table or a calculator with statistical functions (like the `norm.ppf` function in Python's `scipy.stats` library). The z-score for 0.57 is approximately 0.176.\r<BR>\n" );
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document.write( "2. **Use the z-score formula:** The z-score formula is:\r<BR>\n" );
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document.write( "   z = (x - μ) / σ\r<BR>\n" );
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document.write( "   Where:<BR>\n" );
document.write( "   * z is the z-score<BR>\n" );
document.write( "   * x is the value we're looking for (P57)<BR>\n" );
document.write( "   * μ is the mean (199.9)<BR>\n" );
document.write( "   * σ is the standard deviation (82)\r<BR>\n" );
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document.write( "3. **Solve for x (P57):**\r<BR>\n" );
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document.write( "   0.176 = (x - 199.9) / 82\r<BR>\n" );
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document.write( "   Multiply both sides by 82:\r<BR>\n" );
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document.write( "   14.432 = x - 199.9\r<BR>\n" );
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document.write( "   Add 199.9 to both sides:\r<BR>\n" );
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document.write( "   x = 214.332\r<BR>\n" );
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document.write( "Therefore, P57 ≈ 214.3320<BR>\n" );
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