document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1172346>Question 1172346</A>: <I><SPAN STYLE=\"word-break: break-all;\">  To qualify for a police academy, applicants are given a test of physical fitness. The scores are normally distributed with a mean of 64 and a standard deviation of 9. If only the top 20% of the applicants are selected, find the cutoff score.<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 #797361 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=797361\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> mean is 64<BR>\n" );
document.write( "standard deviation is 9<BR>\n" );
document.write( "to be in the top 20%, the z-score needs to be greater than 80%.<BR>\n" );
document.write( "that would be the area to the left of the z-score.<BR>\n" );
document.write( "if you look in the z-score table for an area of .8 to the left of the indicated z-score, you will find:<BR>\n" );
document.write( "z-score is between .84 and .85.<BR>\n" );
document.write( "interpolation yields .8416129032<BR>\n" );
document.write( "raw score yields 71.57451613<BR>\n" );
document.write( "that would be the score that he needs to be better than 80% of the other applicants.<BR>\n" );
document.write( "you could save yourself a lot of grief by using a z-score calculator.<BR>\n" );
document.write( "in that calculator, .....<BR>\n" );
document.write( "provide the mean of 64.<BR>\n" );
document.write( "provide the standard deviation of 9.<BR>\n" );
document.write( "the calculator should give you the raw score indicated above, perhaps rounded a bit differently.<BR>\n" );
document.write( "here's a picture from one of the easiest of them to use.\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" src = \"http://theo.x10hosting.com/2020/122602.jpg\" >\r<BR>\n" );
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document.write( "this calculator can be 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( "if you do not have access to a calculator and must use the z-score tables, then  the procedure would be as shown below:\r<BR>\n" );
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document.write( "look in the z-score for an area to the left of the z-score closest to .8.<BR>\n" );
document.write( "one value above.<BR>\n" );
document.write( "one value below.\r<BR>\n" );
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document.write( "i found:<BR>\n" );
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document.write( "             z-score             area to the left\r\n" );
document.write( "                .84                  .79955\r\n" );
document.write( "                .85                  .80234\r\n" );
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document.write( "difference      .01                  .00279\r\n" );
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document.write( "the area to the left of the z-score you want is .8.\r<BR>\n" );
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document.write( ".8 - .79955 = .00045\r<BR>\n" );
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document.write( "your ratio is:\r<BR>\n" );
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document.write( "x / (.01) = .00045 / .00279\r<BR>\n" );
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document.write( "solve for x to get:\r<BR>\n" );
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document.write( "x = .00045 /.00279 * .01 = .0016129032.\r<BR>\n" );
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document.write( "add this to the z-score of .84 to get:<BR>\n" );
document.write( "your z-score is .84 + .0016129032 = .8416129032.\r<BR>\n" );
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document.write( "use  the z-score formula to find the raw score.\r<BR>\n" );
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document.write( "z = (x - m) / s becomes .8416129032 = x - 64) / 9<BR>\n" );
document.write( "solve for x to get:<BR>\n" );
document.write( "x = 9 * .8416129032 + 64 = 71.57451612\r<BR>\n" );
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document.write( "if you round this to 3 decimal places, it will be 71.575, same as the calculator provided.\r<BR>\n" );
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document.write( "the x that i used for interpolation is not the same x that i used to find the raw score.<BR>\n" );
document.write( "same variable name used in two different procedures is acceptable.<BR>\n" );
document.write( "they are not referring to the same thing.\r<BR>\n" );
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document.write( "manual interpolation can be tricky, but it does work reasonable well.<BR>\n" );
document.write( "with so many calculators that do z-scores and raw scores around, it's usually not necessary.\r<BR>\n" );
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