document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1168870>Question 1168870</A>: <I><SPAN STYLE=\"word-break: break-all;\"> The length of human pregnancies from conception to birth varies according to an approximately normal distribution with a mean of 266 days and a standard deviation of 16 days. Use the 68-95-99.7 Rule to answer the following questions.\r<BR>\n" );
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document.write( "(a) What percentage of pregnancies last between 250 and 282 days?\r<BR>\n" );
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document.write( "(b) What percentage of pregnancies last fewer than 250 days?\r<BR>\n" );
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document.write( "(c) What percentage of pregnancies last between 266 and 298 days?\r<BR>\n" );
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document.write( "(d) What percentage of pregnancies last more than 298 days?\r<BR>\n" );
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document.write( "(e) What percentage of pregnancies last between 234 and 282 days?<BR>\n" );
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document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #851488 by </B><A HREF=/tutors/aboutme.mpl?userid=CPhill><B>CPhill(1959)</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=851488\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> Let's use the 68-95-99.7 Rule to answer these questions.\r<BR>\n" );
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document.write( "**Understanding the 68-95-99.7 Rule**\r<BR>\n" );
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document.write( "* In a normal distribution:<BR>\n" );
document.write( "    * Approximately 68% of the data falls within 1 standard deviation of the mean.<BR>\n" );
document.write( "    * Approximately 95% of the data falls within 2 standard deviations of the mean.<BR>\n" );
document.write( "    * Approximately 99.7% of the data falls within 3 standard deviations of the mean.\r<BR>\n" );
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document.write( "**Given Information**\r<BR>\n" );
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document.write( "* Mean (µ) = 266 days<BR>\n" );
document.write( "* Standard deviation (σ) = 16 days\r<BR>\n" );
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document.write( "**Calculations**\r<BR>\n" );
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document.write( "* µ - σ = 266 - 16 = 250<BR>\n" );
document.write( "* µ + σ = 266 + 16 = 282<BR>\n" );
document.write( "* µ - 2σ = 266 - 32 = 234<BR>\n" );
document.write( "* µ + 2σ = 266 + 32 = 298<BR>\n" );
document.write( "* µ - 3σ = 266 - 48 = 218<BR>\n" );
document.write( "* µ + 3σ = 266 + 48 = 314\r<BR>\n" );
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document.write( "**(a) What percentage of pregnancies last between 250 and 282 days?**\r<BR>\n" );
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document.write( "* This is the range of µ - σ to µ + σ, which is 1 standard deviation from the mean.<BR>\n" );
document.write( "* Therefore, approximately 68% of pregnancies last between 250 and 282 days.\r<BR>\n" );
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document.write( "**(b) What percentage of pregnancies last fewer than 250 days?**\r<BR>\n" );
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document.write( "* 250 days is µ - σ.<BR>\n" );
document.write( "* Since 68% of pregnancies fall within 1 standard deviation of the mean, 32% fall outside of it (100% - 68% = 32%).<BR>\n" );
document.write( "* Because the normal distribution is symmetrical, half of the 32% falls below 250 days.<BR>\n" );
document.write( "* 32%/2 = 16%<BR>\n" );
document.write( "* Therefore, approximately 16% of pregnancies last fewer than 250 days.\r<BR>\n" );
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document.write( "**(c) What percentage of pregnancies last between 266 and 298 days?**\r<BR>\n" );
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document.write( "* 266 is the mean (µ).<BR>\n" );
document.write( "* 298 is µ + 2σ.<BR>\n" );
document.write( "* 95% of the data falls within 2 standard deviations. That means 47.5% of the data falls between the mean, and two standard deviations above the mean.<BR>\n" );
document.write( "* Therefore, approximately 47.5% of pregnancies last between 266 and 298 days.\r<BR>\n" );
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document.write( "**(d) What percentage of pregnancies last more than 298 days?**\r<BR>\n" );
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document.write( "* 298 is µ + 2σ.<BR>\n" );
document.write( "* 95% of the data falls within 2 standard deviations. That leaves 5% outside of that range.<BR>\n" );
document.write( "* Because the normal distribution is symmetrical, half of the 5% falls above 298 days.<BR>\n" );
document.write( "* 5%/2 = 2.5%<BR>\n" );
document.write( "* Therefore, approximately 2.5% of pregnancies last more than 298 days.\r<BR>\n" );
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document.write( "**(e) What percentage of pregnancies last between 234 and 282 days?**\r<BR>\n" );
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document.write( "* 234 is µ - 2σ.<BR>\n" );
document.write( "* 282 is µ + σ.<BR>\n" );
document.write( "* The range between 234 and 266 is 47.5% of the data.<BR>\n" );
document.write( "* The range between 266 and 282 is 34% of the data.<BR>\n" );
document.write( "* 47.5% + 34% = 81.5%<BR>\n" );
document.write( "* Therefore, approximately 81.5% of pregnancies last between 234 and 282 days.<BR>\n" );
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