document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1178774>Question 1178774</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Let x denote the time taken to run a road race. Suppose x is approximately normally<BR>\n" );
document.write( "distributed with a mean of 190 minutes and a standard deviation of 21 minutes. If<BR>\n" );
document.write( "one runner is selected at random, what is the probability that this runner will<BR>\n" );
document.write( "complete this road race in:<BR>\n" );
document.write( "a. less than 160 minutes? [3 marks]<BR>\n" );
document.write( "b. greater than 210 minutes? [5 marks]<BR>\n" );
document.write( "c. 215 to 245 minutes? [7 minutes] </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #850299 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=850299\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> Let's solve 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( "* X ~ N(190, 21²) (X is normally distributed with mean μ = 190 minutes and standard deviation σ = 21 minutes)\r<BR>\n" );
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document.write( "**a. Less than 160 minutes?**\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 = (160 - 190) / 21<BR>\n" );
document.write( "    * z = -30 / 21<BR>\n" );
document.write( "    * z ≈ -1.4286\r<BR>\n" );
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document.write( "2.  **Find the probability:**<BR>\n" );
document.write( "    * P(X < 160) = P(z < -1.4286)<BR>\n" );
document.write( "    * Using a z-table or calculator, P(z < -1.43) ≈ 0.0764\r<BR>\n" );
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document.write( "**b. Greater than 210 minutes?**\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 = (210 - 190) / 21<BR>\n" );
document.write( "    * z = 20 / 21<BR>\n" );
document.write( "    * z ≈ 0.9524\r<BR>\n" );
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document.write( "2.  **Find the probability:**<BR>\n" );
document.write( "    * P(X > 210) = P(z > 0.9524)<BR>\n" );
document.write( "    * Using a z-table or calculator, P(z > 0.95) ≈ 0.1711\r<BR>\n" );
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document.write( "**c. 215 to 245 minutes?**\r<BR>\n" );
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document.write( "1.  **Calculate the z-scores:**<BR>\n" );
document.write( "    * z₁ = (215 - 190) / 21<BR>\n" );
document.write( "    * z₁ = 25 / 21<BR>\n" );
document.write( "    * z₁ ≈ 1.1905<BR>\n" );
document.write( "    * z₂ = (245 - 190) / 21<BR>\n" );
document.write( "    * z₂ = 55 / 21<BR>\n" );
document.write( "    * z₂ ≈ 2.6190\r<BR>\n" );
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document.write( "2.  **Find the probability:**<BR>\n" );
document.write( "    * P(215 < X < 245) = P(1.1905 < z < 2.6190)<BR>\n" );
document.write( "    * Using a z-table or calculator:<BR>\n" );
document.write( "        * P(z < 2.6190) ≈ 0.9956<BR>\n" );
document.write( "        * P(z < 1.1905) ≈ 0.8830<BR>\n" );
document.write( "        * P(1.1905 < z < 2.6190) = P(z < 2.6190) - P(z < 1.1905) ≈ 0.9956 - 0.8830 ≈ 0.1126<BR>\n" );
document.write( " </SPAN>\n" );
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