document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=854240>Question 854240</A>: <I><SPAN STYLE=\"word-break: break-all;\"> A statistical analysis of a telephone calls made by a company indicates that the length of normally distributed, with mean( u=260 seconds) and variance =40 seconds.\r<BR>\n" );
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document.write( "(a) what is the probability that a call lasted between 190 and 340 seconds?<BR>\n" );
document.write( "(b) what is the length of a call if only 1% of calls are shorter ?<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 #514614 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=514614\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> mean = 260<BR>\n" );
document.write( "variance = 40<BR>\n" );
document.write( "standard deviation = square root of variance = square root (40) = 6.324555 rounded to 6 decimal places.\r<BR>\n" );
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document.write( "(a) what is the probability that a call lasted between 190 and 340 seconds?<BR>\n" );
document.write( "(b) what is the length of a call if only 1% of calls are shorter ?\r<BR>\n" );
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document.write( "z-score = (x-m)/s\r<BR>\n" );
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document.write( "x is the 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( "you are looking at 2 z-scores and the probability that the call will be between them.\r<BR>\n" );
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document.write( "x1 = 190 seconds<BR>\n" );
document.write( "x2 = 340 seconds\r<BR>\n" );
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document.write( "z1 = (x1 - m) / s<BR>\n" );
document.write( "x1 = 190<BR>\n" );
document.write( "m = 260<BR>\n" );
document.write( "s = 6.324555\r<BR>\n" );
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document.write( "z1 = (190 - 260) / 6.324555\r<BR>\n" );
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document.write( "this makes z1 = -11.06797237\r<BR>\n" );
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document.write( "that looks awfully big z.\r<BR>\n" );
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document.write( "if you meant standard deviation = 40 then the results would look more normal.\r<BR>\n" );
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document.write( "i'll assume you meant standard deviation = 40 and work the problem that way because variance equal 40 just doesn't make much sense.\r<BR>\n" );
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document.write( "standard deviation is equal to the square root of the variance so variance of 40 gives us a standard deviation that's just too small for this type of problem.\r<BR>\n" );
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document.write( "i suspect standard deviation is 40 would be more in line with what it should be rather than variance = 40.\r<BR>\n" );
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document.write( "assuming that the standard deviation is 40, the results would be as follows:\r<BR>\n" );
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document.write( "z1 = (190 - 260) / 40 = -1.75<BR>\n" );
document.write( "z2 = (340 - 260) / 40 = 2\r<BR>\n" );
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document.write( "you would look up the z-score of -1.75 and the z-score of 2 and you would then subtract the area to the left of these z-scores to get the area in between and you would get:\r<BR>\n" );
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document.write( "p(z2-z1) = .9772 - .0401 = .9371\r<BR>\n" );
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document.write( "there is a 93.71% probability that a the z-score you get will be between a z-score of -1.75 and 2.\r<BR>\n" );
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document.write( "put into words of real scores, there is a 93.71% probability that the duration of the call will be between 190 and 340 seconds.\r<BR>\n" );
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document.write( "the table i used is at the following link:\r<BR>\n" );
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document.write( "<a  rel=nofollow HREF= \"http://lilt.ilstu.edu/dasacke/eco148/ztable.htm\" target = \"_blank\">http://lilt.ilstu.edu/dasacke/eco148/ztable.htm</a>\r<BR>\n" );
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document.write( "if you want to find the duration of the call where only 1% of the calls are shorter, you would use this table again and find the z-score that gives you an area to the left of it that is closest to .01\r<BR>\n" );
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document.write( "in that table, i found that an area to the left of the z-score that gave an area to the left of it closest to .01 was equal to -2.33.\r<BR>\n" );
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document.write( "since z-score = (x-m)/s, it follows that x = s * z-score + m.\r<BR>\n" );
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document.write( "assuming s = 40 and m = 260 and z = -2.33, this formula  yields:\r<BR>\n" );
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document.write( "x = 40 * -2.33 + 260 which is equal to 166.8\r<BR>\n" );
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document.write( "we can back track with that value for x and see if we did it right.\r<BR>\n" );
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document.write( "z = (x - m) / s which becomes z = (166.8 - 260) / 40 which becomes z = -2.33.\r<BR>\n" );
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document.write( "we then look up in the table for the area to the left of that and we get .0099, so we confirmed that we did it correctly.\r<BR>\n" );
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document.write( "that'a about the best that we can do given the accuracy of the tables and it's close enough for the type of problem that we're dealing with.\r<BR>\n" );
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document.write( "i used my TI-84 calculator to find the z-score for an area to the left of it of .01 and i got z = -2.326347877.\r<BR>\n" );
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document.write( "round that to 2 decimal places and you get -2.33 so we did pretty good.\r<BR>\n" );
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