Question 1117687: 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
a.greater than 50 days
b.less than 30 days
c.between 30 and 60 days
d.greater than 90 days
Answer by Theo(13342)   (Show Source):
You can put this solution on YOUR website! assuming a reasonablhy normal distribution, then you would calculate as follows:
the mean is 60
the standard deviation is 15.
you would find the z-score and then determine the probability from the normal distribution table, or a normal distribution calculator.
the formuloa for z-wscore is:
z = (x-m)/s
z is the z-score
x is the raw score you are compqring to the mean.
m is the mean.
s is the standard deviation.
if the z-score is 0, your raw score is equal to the mean.
if the z-score is positive, your raw score is greater than the mean.
if the z-score is negative, your raw score is less than the mean.
the z-score tells you how many standard deviations your raw score is greater than or less than the mean.
this will become apparent as we go through the test cases.
your questions are:
a.greater than 50 days
x = 50
m = 60
s = 15
z = (x-m)/s = (50 - 60) / 15 = -10/15 = -.67 rounded to 2 decimal places.
this means that your raw score is .67 standard deviations below the mean.
your would look up a z-score of -.67 in the z-score table.
it will tell you that the probability of getting a z-score less than -.67 is equal to .2514.
the z-score table tells you the probability of less than.
that's the area under the normal distribution curve to the left of your z-score.
to find the probability of greater than, you would take 1 minus the probability of less than.
in this case, that would be 1 - .2514 = .7486.
since you are looking for the probability that your raw score is greater than 50, then your answer would be .7486.
b.less than 30 days
z = (x-m)/s becomews z = (30-60)/15 which becomes z = -30/15 which becomes z = -2.
this says that your raw score is 2 standard deviations below the mean.
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.
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.
c.between 30 and 60 days
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.
that gives you the area in between.
z-score of 30 = (30 - 60) / 15 = -2.
z-score of 60 = (60 - 60) / 15 = 0.
area to the left of a z-score of 0 is .5
area to the left of a z-score of -2 is .0228
area in between is .5 - .0228 = .4772.
d.greater than 90 days
z = (90 - 60) / 15 = 30 / 15 = 2
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.
note that the normal distribution is symmetrical about the mean.
the mean is .5
the median is .5
the mode is .5
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.
for xample:
z-score is 2 or -2.
area to the left of a z-score of -2 is equal to .0228
area to the right of a z-score of 2 is equal to .0228.
area to the right of a z-score of -2 is equal to .9772.
area to the left of a z-score of 2 is equal to .9772.
there is a online calculator that is very easy to use and also give you a graphical representation of your date.
that calculator can be found at http://davidmlane.com/hyperstat/z_table.html
you can use it with raw scores or with z-scores.
the answers will be the same.
your problems are done again using this normal distribution calculator.
first is using the z-score, second is using the raw score.
a.greater than 50 days
the difference between the probabilities here is due to rounding.
the more exact z-score is -.666666666......
it was rounded to -.67.
if it was shown as -.66666666, it would be the same results as using the raw score result.
b.less than 30 days
c.between 30 and 60 days
d.greater than 90 days
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