document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1179878>Question 1179878</A>: <I><SPAN STYLE=\"word-break: break-all;\"> The daily temperature in February is normally distributed with a mean of 25 degrees and a standard deviation of 12 degrees. What percentage of the time will this temperature be:\r<BR>\n" );
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document.write( "A. above freezing? (32 degrees or more)\r<BR>\n" );
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document.write( "B. below zero?\r<BR>\n" );
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document.write( "c. between 35 degrees and 45 degrees?\r<BR>\n" );
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document.write( "Explain your answers.  </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #850160 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=850160\">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:\r<BR>\n" );
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document.write( "Mean (μ) = 25 degrees<BR>\n" );
document.write( "Standard deviation (σ) = 12 degrees<BR>\n" );
document.write( "Distribution: Normal<BR>\n" );
document.write( "A. Above Freezing (32 degrees or more):\r<BR>\n" );
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document.write( "Calculate the z-score:\r<BR>\n" );
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document.write( "z = (x - μ) / σ<BR>\n" );
document.write( "z = (32 - 25) / 12<BR>\n" );
document.write( "z = 7 / 12 ≈ 0.58<BR>\n" );
document.write( "Find the probability:\r<BR>\n" );
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document.write( "We want P(X ≥ 32), which is the same as P(Z ≥ 0.58).<BR>\n" );
document.write( "Using a z-table or calculator, find P(Z < 0.58) ≈ 0.7190.<BR>\n" );
document.write( "P(Z ≥ 0.58) = 1 - P(Z < 0.58) = 1 - 0.7190 = 0.2810<BR>\n" );
document.write( "Convert to percentage:\r<BR>\n" );
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document.write( "0.2810 * 100% = 28.10%<BR>\n" );
document.write( "Explanation: There's about a 28.10% chance that the daily temperature will be above freezing (32 degrees or more).\r<BR>\n" );
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document.write( "B. Below Zero:\r<BR>\n" );
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document.write( "Calculate the z-score:\r<BR>\n" );
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document.write( "z = (x - μ) / σ<BR>\n" );
document.write( "z = (0 - 25) / 12<BR>\n" );
document.write( "z = -25 / 12 ≈ -2.08<BR>\n" );
document.write( "Find the probability:\r<BR>\n" );
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document.write( "We want P(X < 0), which is the same as P(Z < -2.08).<BR>\n" );
document.write( "Using a z-table or calculator, find P(Z < -2.08) ≈ 0.0188.<BR>\n" );
document.write( "Convert to percentage:\r<BR>\n" );
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document.write( "0.0188 * 100% = 1.88%<BR>\n" );
document.write( "Explanation: There's about a 1.88% chance that the daily temperature will be below zero.\r<BR>\n" );
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document.write( "C. Between 35 degrees and 45 degrees:\r<BR>\n" );
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document.write( "Calculate the z-scores:\r<BR>\n" );
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document.write( "For 35 degrees:<BR>\n" );
document.write( "z1 = (35 - 25) / 12 = 10 / 12 ≈ 0.83<BR>\n" );
document.write( "For 45 degrees:<BR>\n" );
document.write( "z2 = (45 - 25) / 12 = 20 / 12 ≈ 1.67<BR>\n" );
document.write( "Find the probability:\r<BR>\n" );
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document.write( "We want P(35 ≤ X ≤ 45), which is the same as P(0.83 ≤ Z ≤ 1.67).<BR>\n" );
document.write( "P(0.83 ≤ Z ≤ 1.67) = P(Z ≤ 1.67) - P(Z ≤ 0.83)<BR>\n" );
document.write( "Using a z-table or calculator:<BR>\n" );
document.write( "P(Z ≤ 1.67) ≈ 0.9525<BR>\n" );
document.write( "P(Z ≤ 0.83) ≈ 0.7967<BR>\n" );
document.write( "P(0.83 ≤ Z ≤ 1.67) = 0.9525 - 0.7967 = 0.1558<BR>\n" );
document.write( "Convert to percentage:\r<BR>\n" );
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document.write( "0.1558 * 100% = 15.58%<BR>\n" );
document.write( "Explanation: There's about a 15.58% chance that the daily temperature will be between 35 and 45 degrees.\r<BR>\n" );
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document.write( "Answers:\r<BR>\n" );
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document.write( "A. 28.10%<BR>\n" );
document.write( "B. 1.88%<BR>\n" );
document.write( "C. 15.58% </SPAN>\n" );
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