document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1178781>Question 1178781</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Private nonprofit four-year colleges charge, on average, $26,423 per year in tuition and fees. The standard deviation is $7,434. Assume the distribution is normal. Let X be the cost for a randomly selected college. Round all answers to 4 decimal places where possible.\r<BR>\n" );
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document.write( "a. What is the distribution of X? X ~ N( ,)\r<BR>\n" );
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document.write( "b. Find the probability that a randomly selected Private nonprofit four-year college will cost less than 23,012 per year. <BR>\n" );
document.write( " (Round z-score up to 2 decimal places.)\r<BR>\n" );
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document.write( "c. Find the 80th percentile for this distribution. $<BR>\n" );
document.write( " (Round to the nearest dollar.) </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #850295 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=850295\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> Here's how to solve this problem:\r<BR>\n" );
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document.write( "**a. What is the distribution of X?**\r<BR>\n" );
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document.write( "* X follows a normal distribution with a mean (μ) of $26,423 and a standard deviation (σ) of $7,434.<BR>\n" );
document.write( "* Therefore, X ~ N(26423, 7434)\r<BR>\n" );
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document.write( "**b. Find the probability that a randomly selected Private nonprofit four-year college will cost less than $23,012 per year.**\r<BR>\n" );
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document.write( "1.  **Calculate the z-score:**<BR>\n" );
document.write( "    * z = (X - μ) / σ<BR>\n" );
document.write( "    * z = (23012 - 26423) / 7434<BR>\n" );
document.write( "    * z = -3411 / 7434<BR>\n" );
document.write( "    * z ≈ -0.4588<BR>\n" );
document.write( "    * Rounding up to 2 decimal places, z = -0.46\r<BR>\n" );
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document.write( "2.  **Find the probability:**<BR>\n" );
document.write( "    * P(X < 23012) = P(z < -0.46)<BR>\n" );
document.write( "    * Using a z-table or calculator, P(z < -0.46) ≈ 0.3228\r<BR>\n" );
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document.write( "**c. Find the 80th percentile for this distribution.**\r<BR>\n" );
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document.write( "1.  **Find the z-score for the 80th percentile:**<BR>\n" );
document.write( "    * Using a z-table or calculator, the z-score corresponding to the 80th percentile is approximately 0.84.\r<BR>\n" );
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document.write( "2.  **Calculate the value of X:**<BR>\n" );
document.write( "    * X = μ + z * σ<BR>\n" );
document.write( "    * X = 26423 + 0.84 * 7434<BR>\n" );
document.write( "    * X = 26423 + 6244.56<BR>\n" );
document.write( "    * X = 32667.56\r<BR>\n" );
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document.write( "3.  **Round to the nearest dollar:**<BR>\n" );
document.write( "    * X ≈ $32,668<BR>\n" );
document.write( " </SPAN>\n" );
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