document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1191361>Question 1191361</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Suppose SPS professors have an average life normally distributed of 73 years, with a population standard deviation of 6 years.  <BR>\n" );
document.write( "a) What percent of SPS professors will live less than 60 years?<BR>\n" );
document.write( "b) What proportion of SPS professors will live between 85 and 90 years?<BR>\n" );
document.write( "c) Calculate the 30th percentile.<BR>\n" );
document.write( "d) Calculate the 97th percentile.<BR>\n" );
document.write( "e) What percent of SPS professors will make it past the age of 75?<BR>\n" );
document.write( "  </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #849221 by </B><A HREF=/tutors/aboutme.mpl?userid=CPhill><B>CPhill(2189)</B></A><img src=\"/images/stars/stars-13.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=849221\">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) Percent living less than 60 years:**\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 = (60 - 73) / 6<BR>\n" );
document.write( "   z = -2.17\r<BR>\n" );
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document.write( "2. **Find the probability:** Use a z-table or calculator to find the area to the *left* of z = -2.17. This gives the probability of living less than 60 years. P(z < -2.17) ≈ 0.015 or 1.5%\r<BR>\n" );
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document.write( "**b) Proportion living between 85 and 90 years:**\r<BR>\n" );
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document.write( "1. **Calculate the z-scores:**<BR>\n" );
document.write( "   z₁ = (85 - 73) / 6 = 2<BR>\n" );
document.write( "   z₂ = (90 - 73) / 6 = 2.83\r<BR>\n" );
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document.write( "2. **Find the probabilities:** Use a z-table or calculator.<BR>\n" );
document.write( "   P(z < 2) ≈ 0.9772<BR>\n" );
document.write( "   P(z < 2.83) ≈ 0.9977\r<BR>\n" );
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document.write( "3. **Find the proportion between the two ages:**<BR>\n" );
document.write( "   P(2 < z < 2.83) = P(z < 2.83) - P(z < 2) = 0.9977 - 0.9772 ≈ 0.0205\r<BR>\n" );
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document.write( "**c) 30th percentile:**\r<BR>\n" );
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document.write( "1. **Find the z-score:** The 30th percentile corresponds to a cumulative probability of 0.30. Look up the z-score closest to 0.30 in the z-table; the z-score is approximately -0.52.\r<BR>\n" );
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document.write( "2. **Use the z-score formula:**<BR>\n" );
document.write( "   x = μ + zσ<BR>\n" );
document.write( "   x = 73 + (-0.52 * 6)<BR>\n" );
document.write( "   x ≈ 69.88 years\r<BR>\n" );
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document.write( "**d) 97th percentile:**\r<BR>\n" );
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document.write( "1. **Find the z-score:**  The 97th percentile corresponds to a cumulative probability of 0.97. The z-score is approximately 1.88.\r<BR>\n" );
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document.write( "2. **Use the z-score formula:**<BR>\n" );
document.write( "   x = μ + zσ<BR>\n" );
document.write( "   x = 73 + (1.88 * 6)<BR>\n" );
document.write( "   x ≈ 84.28 years\r<BR>\n" );
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document.write( "**e) Percent making it past 75:**\r<BR>\n" );
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document.write( "1. **Calculate the z-score:**<BR>\n" );
document.write( "   z = (75 - 73) / 6<BR>\n" );
document.write( "   z = 0.33\r<BR>\n" );
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document.write( "2. **Find the probability:** Use a z-table or calculator to find the area to the *right* of z = 0.33. This is 1 - P(z < 0.33). P(z < 0.33) ≈ 0.6293. So, 1 - 0.6293 ≈ 0.3707 or 37.07%<BR>\n" );
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