document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1117687>Question 1117687</A>: <I><SPAN STYLE=\"word-break: break-all;\"> suppose the average length of stay in a chronic disease hospital of a certain type of patient is 60 days with a standard deviation of 15.if it is reasonable to assume an approximately normal distribution of length of stay, find the probability that a randomly selected patient from this group will have a length of stay<BR>\n" );
document.write( "a.greater than 50 days<BR>\n" );
document.write( "b.less than 30 days<BR>\n" );
document.write( "c.between 30 and 60 days<BR>\n" );
document.write( "d.greater than 90 days </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #732984 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=732984\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> assuming a reasonablhy normal distribution, then you would calculate as follows:\r<BR>\n" );
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document.write( "the mean is 60<BR>\n" );
document.write( "the standard deviation is 15.\r<BR>\n" );
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document.write( "you would find the z-score and then determine the probability from the normal distribution table, or a normal distribution calculator.\r<BR>\n" );
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document.write( "the formuloa for z-wscore is:\r<BR>\n" );
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document.write( "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 you are compqring to the mean.<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( "if the z-score is 0, your raw score is equal to the mean.\r<BR>\n" );
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document.write( "if the z-score is positive, your raw score is greater than the mean.\r<BR>\n" );
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document.write( "if the z-score is negative, your raw score is less than the mean.\r<BR>\n" );
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document.write( "the z-score tells you how many standard deviations your raw score is greater than or less than the mean.\r<BR>\n" );
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document.write( "this will become apparent as we go through the test cases.\r<BR>\n" );
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document.write( "your questions are:\r<BR>\n" );
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document.write( "a.greater than 50 days\r<BR>\n" );
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document.write( "x = 50<BR>\n" );
document.write( "m = 60<BR>\n" );
document.write( "s = 15\r<BR>\n" );
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document.write( "z = (x-m)/s = (50 - 60) / 15 = -10/15 = -.67 rounded to 2 decimal places.\r<BR>\n" );
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document.write( "this means that your raw score is .67 standard deviations below the mean.\r<BR>\n" );
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document.write( "your would look up a z-score of -.67 in the z-score table.<BR>\n" );
document.write( "it will tell you that the probability of getting a z-score less than -.67 is equal to .2514.\r<BR>\n" );
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document.write( "the z-score table tells you the probability of less than.<BR>\n" );
document.write( "that's the area under the normal distribution curve to the left of your z-score.\r<BR>\n" );
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document.write( "to find the probability of greater than, you would take 1 minus the probability of less than.\r<BR>\n" );
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document.write( "in this case, that would be 1 - .2514 = .7486.\r<BR>\n" );
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document.write( "since you are looking for the probability that your raw score is greater than 50, then your answer would be .7486.\r<BR>\n" );
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document.write( "b.less than 30 days\r<BR>\n" );
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document.write( "z = (x-m)/s becomews z = (30-60)/15 which becomes z = -30/15 which becomes z = -2.\r<BR>\n" );
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document.write( "this says that your raw score is 2 standard deviations below the mean.\r<BR>\n" );
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document.write( "you look up a z-score of -2 in the z-score table and it tells you that the probability of getting a z-score less than -2 is equal to .0228, given that the mean is 60 and the standard deviation is 15.\r<BR>\n" );
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document.write( "no adjustments are necessary, since the z-score table is designed to tell you the probability of getting a z-score less than the indicated z-score.\r<BR>\n" );
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document.write( "c.between 30 and 60 days\r<BR>\n" );
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document.write( "you need to look up both z-scores and then substract the area to the left of the lower z-score from the area to the left of the higher z-score.\r<BR>\n" );
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document.write( "that gives you the area in between.\r<BR>\n" );
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document.write( "z-score of 30 = (30 - 60) / 15 = -2.\r<BR>\n" );
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document.write( "z-score of 60 = (60 - 60) / 15 = 0.\r<BR>\n" );
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document.write( "area to the left of a z-score of 0 is .5<BR>\n" );
document.write( "area to the left of a z-score of -2 is .0228\r<BR>\n" );
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document.write( "area in between is .5 - .0228 = .4772.\r<BR>\n" );
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document.write( "d.greater than 90 days\r<BR>\n" );
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document.write( "z = (90 - 60) / 15 = 30 / 15 = 2\r<BR>\n" );
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document.write( "look up z-score of 2 in the z-score table and it will tell you that the area to the left of that z-score is equal to .9772.\r<BR>\n" );
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document.write( "note that the normal distribution is symmetrical about the mean.\r<BR>\n" );
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document.write( "the mean is .5<BR>\n" );
document.write( "the median is .5<BR>\n" );
document.write( "the mode is .5\r<BR>\n" );
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document.write( "symmetrical means that the difference between a positive z-score from the mean is the same as the difference between the negative value of that same z-score below the mean.\r<BR>\n" );
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document.write( "for xample:\r<BR>\n" );
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document.write( "z-score is 2 or -2.\r<BR>\n" );
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document.write( "area to the left of a z-score of -2 is equal to .0228<BR>\n" );
document.write( "area to the right of a z-score of 2 is equal to .0228.\r<BR>\n" );
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document.write( "area to the right of a z-score of -2 is equal to .9772.<BR>\n" );
document.write( "area to the left of a z-score of 2 is equal to .9772.\r<BR>\n" );
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document.write( "there is a online calculator that is very easy to use and also give you a graphical representation of your date.\r<BR>\n" );
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document.write( "that 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( "you can use it with raw scores or with z-scores.<BR>\n" );
document.write( "the answers will be the same.\r<BR>\n" );
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document.write( "your problems are done again using this normal distribution calculator.<BR>\n" );
document.write( "first is using the z-score, second is using the raw score.\r<BR>\n" );
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document.write( "a.greater than 50 days\r<BR>\n" );
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document.write( "the difference between the probabilities here is due to rounding.\r<BR>\n" );
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document.write( "the more exact z-score is -.666666666......\r<BR>\n" );
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document.write( "it was rounded to -.67.\r<BR>\n" );
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document.write( "if it was shown as -.66666666, it would be the same results as using the raw score result.\r<BR>\n" );
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document.write( "b.less than 30 days\r<BR>\n" );
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document.write( "c.between 30 and 60 days\r<BR>\n" );
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document.write( "d.greater than 90 days\r<BR>\n" );
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document.write( " </SPAN>\n" );
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