document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1138315>Question 1138315</A>: <I><SPAN STYLE=\"word-break: break-all;\"> 99.7% of the scores on an exam are between 60 and 98 points. Assuming the data is normally distributed, what are the mean and standard deviation? </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #756152 by </B><A HREF=/tutors/aboutme.mpl?userid=Theo><B>Theo(13342)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=Theo><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=Theo><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=756152\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> assuming the data is symmetric about the mean, you would do the following.\r<BR>\n" );
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document.write( "the mean would be halfway between 60 and 98.\r<BR>\n" );
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document.write( "take 98 - 60 to get 38 and divide that by 2 to get 19.\r<BR>\n" );
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document.write( "add 19 to 60 and the mean will be 79.\r<BR>\n" );
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document.write( "look up a z-scores that have .997 of the area under the normal distribution curve between them.\r<BR>\n" );
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document.write( "if the area between the z-scores is .997, that means the area outside this interval is 1 - .997 = .003.\r<BR>\n" );
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document.write( "divide that by 2 and the area to the right of the high z-score is .0015 and the area to the left of the low z-score is .0015.\r<BR>\n" );
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document.write( "if the area to the right of the high z-score is .0015, that means the area to the left of the high z-score is 1 - .0015 = .9985.\r<BR>\n" );
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document.write( "the z-score associated with an area of .9885 to the left of it is 2.967737927.\r<BR>\n" );
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document.write( "the z-score associated with an area of .0015 to the left of it is -2.967737927.\r<BR>\n" );
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document.write( "those are the z-scores that have .997 of the area under the normal distribution curve between them.\r<BR>\n" );
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document.write( "this can be seen in the following visual display.\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" src = \"http://theo.x10hosting.com/2019/040531.jpg\" alt=\"$$$\" >\r<BR>\n" );
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document.write( "you would then use the z-score formula to find the standard deviation.\r<BR>\n" );
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document.write( "the z-score formula is z = (x - m) / s\r<BR>\n" );
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document.write( "z is the z-score<BR>\n" );
document.write( "x is the raw score<BR>\n" );
document.write( "m is the mean<BR>\n" );
document.write( "s is the standard deviation\r<BR>\n" );
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document.write( "using the high side z-score, the formula becomes:\r<BR>\n" );
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document.write( "2.967737927 = (98 - 79) / s\r<BR>\n" );
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document.write( "solve for s to get s = 19 / 2.967737927 = 6.40218256.\r<BR>\n" );
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document.write( "that's the standard deviation.\r<BR>\n" );
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document.write( "that standard deviation applies to the high side z-score and the low size z-score because the normal distribution curve is symmetric about the mean.\r<BR>\n" );
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document.write( "to confirm, using the low side z-score the formula becomes:\r<BR>\n" );
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document.write( "-2.967737927 = (60 - 79) / s which becomes -2.967737927 = -19 / s.\r<BR>\n" );
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document.write( "solve for s to get s = -19 / -2.967737927 = the same as we got using the high side z-score.\r<BR>\n" );
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document.write( "you now have the mean and the standard deviation.\r<BR>\n" );
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document.write( "to confirm, use the online distributuion calculator, found at <a  rel=nofollow HREF= \"http://davidmlane.com/hyperstat/z_table.html\" target = \"_blank\">http://davidmlane.com/hyperstat/z_table.html</a>\r<BR>\n" );
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document.write( "this is the same calculator i used above with z-scores, only this time i'm using it with raw scores.\r<BR>\n" );
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document.write( "when using with z-scores, the mean is 0 and the standard deviation is 1.\r<BR>\n" );
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document.write( "when using with raw scores, the mean is the mean of the distribution and the standard deviation is the standard deviation of the distribution.\r<BR>\n" );
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document.write( "the display from that calculator is shown below.\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" src = \"http://theo.x10hosting.com/2019/040532.jpg\" alt=\"$$$\" >\r<BR>\n" );
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document.write( " </SPAN>\n" );
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