document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1171493>Question 1171493</A>: <I><SPAN STYLE=\"word-break: break-all;\"> 	<BR>\n" );
document.write( "In a recent​ year, grade 8 Colorado State public school students taking a mathematics assessment test had a mean score of 290 with a standard deviation of 37.  Assume that the scores are normally distributed.<BR>\n" );
document.write( "​A) Find the probability that a student had a score higher than 320.  Enter your answer as a decimal rounded to 2 places. <BR>\n" );
document.write( "​B) Find the probability that a student had a score between 250 and 300.  Enter your answer as<BR>\n" );
document.write( "a decimal rounded to 2 places. <BR>\n" );
document.write( "​C) If 2000 students are randomly​ selected, how many would you expect to have a test score that is less than​ 280?  Round your answer to the nearest whole number. <BR>\n" );
document.write( "nothing<BR>\n" );
document.write( "​D) What is the score that corresponds to the 99th percentile of all​ scores?  Round your answer to the nearest whole number.  <BR>\n" );
document.write( "nothing<BR>\n" );
document.write( "​E) A random sample of 60 students is drawn from this population.  What is the probability that the mean test score of this group is greater than​ 300?  Enter your answer as a decimal rounded to 2 places </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #850901 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=850901\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> Let's solve each part of this problem step-by-step.\r<BR>\n" );
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document.write( "**Given Information:**\r<BR>\n" );
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document.write( "* Mean (μ) = 290<BR>\n" );
document.write( "* Standard deviation (σ) = 37<BR>\n" );
document.write( "* Scores are normally distributed.\r<BR>\n" );
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document.write( "**A) Probability of a score higher than 320:**\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 = (320 - 290) / 37<BR>\n" );
document.write( "    * z = 30 / 37 ≈ 0.81\r<BR>\n" );
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document.write( "2.  **Find the probability:**<BR>\n" );
document.write( "    * Use a z-table or calculator to find P(Z > 0.81).<BR>\n" );
document.write( "    * P(Z > 0.81) ≈ 0.2090<BR>\n" );
document.write( "    * Rounded to 2 decimal places: 0.21\r<BR>\n" );
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document.write( "**B) Probability of a score between 250 and 300:**\r<BR>\n" );
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document.write( "1.  **Calculate the z-scores:**<BR>\n" );
document.write( "    * z1 = (250 - 290) / 37 = -40 / 37 ≈ -1.08<BR>\n" );
document.write( "    * z2 = (300 - 290) / 37 = 10 / 37 ≈ 0.27\r<BR>\n" );
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document.write( "2.  **Find the probabilities:**<BR>\n" );
document.write( "    * P(Z < 0.27) ≈ 0.6064<BR>\n" );
document.write( "    * P(Z < -1.08) ≈ 0.1401\r<BR>\n" );
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document.write( "3.  **Calculate the probability between:**<BR>\n" );
document.write( "    * P(-1.08 < Z < 0.27) = P(Z < 0.27) - P(Z < -1.08)<BR>\n" );
document.write( "    * = 0.6064 - 0.1401 ≈ 0.4663<BR>\n" );
document.write( "    * Rounded to 2 decimal places: 0.47\r<BR>\n" );
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document.write( "**C) Expected number of students with a score less than 280 (out of 2000):**\r<BR>\n" );
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document.write( "1.  **Calculate the z-score:**<BR>\n" );
document.write( "    * z = (280 - 290) / 37 = -10 / 37 ≈ -0.27\r<BR>\n" );
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document.write( "2.  **Find the probability:**<BR>\n" );
document.write( "    * P(Z < -0.27) ≈ 0.3936\r<BR>\n" );
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document.write( "3.  **Calculate the expected number:**<BR>\n" );
document.write( "    * Expected number = 2000 * 0.3936 ≈ 787.2<BR>\n" );
document.write( "    * Rounded to the nearest whole number: 787\r<BR>\n" );
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document.write( "**D) Score corresponding to the 99th percentile:**\r<BR>\n" );
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document.write( "1.  **Find the z-score:**<BR>\n" );
document.write( "    * Use a z-table or calculator to find the z-score corresponding to the 99th percentile (0.99).<BR>\n" );
document.write( "    * z ≈ 2.33\r<BR>\n" );
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document.write( "2.  **Calculate the score:**<BR>\n" );
document.write( "    * x = μ + zσ<BR>\n" );
document.write( "    * x = 290 + (2.33 * 37)<BR>\n" );
document.write( "    * x = 290 + 86.21 ≈ 376.21<BR>\n" );
document.write( "    * Rounded to the nearest whole number: 376\r<BR>\n" );
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document.write( "**E) Probability that the mean of a sample of 60 students is greater than 300:**\r<BR>\n" );
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document.write( "1.  **Calculate the standard error:**<BR>\n" );
document.write( "    * SE = σ / √n<BR>\n" );
document.write( "    * SE = 37 / √60 ≈ 4.77\r<BR>\n" );
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document.write( "2.  **Calculate the z-score for the sample mean:**<BR>\n" );
document.write( "    * z = (x̄ - μ) / SE<BR>\n" );
document.write( "    * z = (300 - 290) / 4.77 ≈ 2.10\r<BR>\n" );
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document.write( "3.  **Find the probability:**<BR>\n" );
document.write( "    * P(Z > 2.10) ≈ 0.0179<BR>\n" );
document.write( "    * Rounded to 2 decimal places: 0.02<BR>\n" );
document.write( " </SPAN>\n" );
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