document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1203867>Question 1203867</A>: <I><SPAN STYLE=\"word-break: break-all;\"> The mean price paid is $1400 and the standard deviation is $110.\r<BR>\n" );
document.write( "\n" );
document.write( "What is the approximate percentage of buyers who paid between $1290 and $1400?<BR>\n" );
document.write( "%\r<BR>\n" );
document.write( "\n" );
document.write( "What is the approximate percentage of buyers who paid between $1290 and $1510?<BR>\n" );
document.write( "%\r<BR>\n" );
document.write( "\n" );
document.write( "What is the approximate percentage of buyers who paid between $1180 and $1400?<BR>\n" );
document.write( "%\r<BR>\n" );
document.write( "\n" );
document.write( "What is the approximate percentage of buyers who paid less than $1180?<BR>\n" );
document.write( "%\r<BR>\n" );
document.write( "\n" );
document.write( "What is the approximate percentage of buyers who paid less than $1070?<BR>\n" );
document.write( "%\r<BR>\n" );
document.write( "\n" );
document.write( "What is the approximate percentage of buyers who paid between $1070 and $1400?<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 #852254 by </B><A HREF=/tutors/aboutme.mpl?userid=ikleyn><B>ikleyn(53763)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=ikleyn><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=ikleyn><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=852254\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> .  <BR>\n" );
document.write( "The mean price paid is $1400 and the standard deviation is $110.\r<BR>\n" );
document.write( "\n" );
document.write( "(a) What is the approximate percentage of buyers who paid between $1290 and $1400?\r<BR>\n" );
document.write( "\n" );
document.write( "(b) What is the approximate percentage of buyers who paid between $1290 and $1510?\r<BR>\n" );
document.write( "\n" );
document.write( "(c) What is the approximate percentage of buyers who paid between $1180 and $1400?\r<BR>\n" );
document.write( "\n" );
document.write( "(d) What is the approximate percentage of buyers who paid less than $1180?\r<BR>\n" );
document.write( "\n" );
document.write( "(e) What is the approximate percentage of buyers who paid less than $1070?\r<BR>\n" );
document.write( "\n" );
document.write( "(f) What is the approximate percentage of buyers who paid between $1070 and $1400?<BR>\n" );
document.write( "~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;Tutor @Theo solved this problem using calculator.\r<BR>\n" );
document.write( "\n" );
document.write( "&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;He presented the answers referring to the calculator' snapshots.\r<BR>\n" );
document.write( "\n" );
document.write( "&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;But now his video-format is not maintained, so his answers are invisible,\r<BR>\n" );
document.write( "\n" );
document.write( "&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;and, therefore, his solutions are now practically dead and useless.\r<BR>\n" );
document.write( "\n" );
document.write( "&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;Meanwhile, this problem can be solved without using a calculator, because all these cases fall\r<BR>\n" );
document.write( "\n" );
document.write( "&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;under the empirical rules of normal distribution.\r<BR>\n" );
document.write( "\n" );
document.write( "&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;Moreover, from the problem, it is obviously clear that this problem is indented\r<BR>\n" );
document.write( "\n" );
document.write( "&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;to be solved using these empirical rules.\r<BR>\n" );
document.write( "\n" );
document.write( "&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;So, I came to provide a solution in a way as it should be done.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "The Empirical Rule, or the 68-95-99.7 Rule, states that for a normal distribution, <BR>\n" );
document.write( "- approximately 68% of data falls within one standard deviation of the mean, <BR>\n" );
document.write( "- 95% falls within two standard deviations, and <BR>\n" );
document.write( "- 99.7% falls within three standard deviations. <BR>\n" );
document.write( "This rule provides a quick way to understand the spread of data around the mean in symmetrical normal distributions.  \r<BR>\n" );
document.write( "<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "<PRE STYLE=\"white-space: pre-wrap; white-space: -moz-pre-wrap; white-space: -pre-wrap; white-space: -o-pre-wrap; word-wrap: break-word;\">\r\n" );
document.write( "(a) Interval from $1290 to $1400 is from one standard deviation left of the mean to the mean,\r\n" );
document.write( "    so it is half of the complete interval \"one standard deviation from the mean\".\r\n" );
document.write( "\r\n" );
document.write( "    Therefore, due to the empirical rule and the symmetry of standard distribution, the answer in this case \r\n" );
document.write( "    is half of 68%, i.e. 34%, approximately.\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "(b) Interval from $1290 to $1510 is precisely \"one standard deviation from the mean\" in both directions.\r\n" );
document.write( "\r\n" );
document.write( "    Therefore, due to the empirical rule, the answer in this case is half of 68%, \r\n" );
document.write( "    i.e. 68%, approximately.\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "(c) Interval from $1180 to $1400 is from two standard deviations left of the mean to the mean,\r\n" );
document.write( "    so it is half of the complete interval \"two standard deviation from the mean\".\r\n" );
document.write( "\r\n" );
document.write( "    Therefore, due to the empirical rule and symmetry of standard distribution, the answer in this case \r\n" );
document.write( "    is half of 95%, i.e. 47.5%, approximately.\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "(d)  In this case, we want to find the area under the normal distribution curve on the left  \r\n" );
document.write( "     from two standard deviations from the mean.\r\n" );
document.write( "\r\n" );
document.write( "     So, we subtract half of 95% from 50%, which corresponds to the mean.\r\n" );
document.write( "     We get then the answer 50% - 47.5% = 2.5%.\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "(e)  In this case, $1070 is in 3 standard deviation from the mean.\r\n" );
document.write( "     So, we want to find the area under the normal distribution curve on the left from three standard \r\n" );
document.write( "     deviations from the mean.\r\n" );
document.write( "\r\n" );
document.write( "     Therefore, the answer for case (d) is 50% minus half of 99.7%, i.e. 50% - 49.85% = 0.15%, approximately.\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "(f)  The interval from $1070 t0 $1400 is three standard deviation on the left from the mean.\r\n" );
document.write( "\r\n" );
document.write( "     So, the answer in this case is half of 99.7%, i.e. about 49.85%, approximately.\r\n" );
document.write( "</PRE>\r<BR>\n" );
document.write( "\n" );
document.write( "Thus, all questions are answered and the problem is solved completely.<BR>\n" );
document.write( "It is done mentally, using the empirical rules for normal distribution, <BR>\n" );
document.write( "precisely in a way as it should be done and as it was expected, due to the design of this problem.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "//////////////////////////////////////////\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "There is another (= one more) reason why I produced and placed my solution here.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "@Theo' posts used visual plots to support his solutions.<BR>\n" );
document.write( "These plots were integral inseparable part of his solutions.<BR>\n" );
document.write( "But some time ago, Theo left this forum and stopped supporting web-site with his plots.<BR>\n" );
document.write( "As a result, you see now some colored spots in his posts, where his plots should be.<BR>\n" );
document.write( "Due to this reason, @Theo's post lost their educational meaning and value.<BR>\n" );
document.write( "Therefore, I create my posts with my own mathematical solutions<BR>\n" );
document.write( "to replace @Theo' solutions and provide meaningful mathematical content.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( " </SPAN>\n" );
document.write( "</TD></TR></TABLE>\n" );