STAT 366 Spring 10 Group Work #4 3/08/10
1) You are a nurse at a community health clinic, and your clinic sees, on average,
about 25 patients per day. On any given day, the number of patients needing
stitches is distributed as follows: 0 patients 10% of the time, 1 patient 10% of the
time, 2 patients 20% of the time, 3 patients 20% of the time, 4 patients 20% of the
time , 5 patients 10% of the time and 6 patients 10% of the time. Each of these
patients will need a special “laceration” package used for sterilizing and stitching a
wound.
a) Make a box model for the distribution of how many laceration packs you will
need each day. What are the mean and SD of the box?
Ans: The box must reflect the percentages of the outcomes given, so the box looks like
[0,1,2,2,3,3,4,4,5,6].
AVG(box)= (The sum of all the outcomes in the box)/(# of outcomes in the box)
= (0+1+2+2+3+3+4+4+5+6)/10 = 3
SD(box)=Sqrt{[(0-3)2 +(1-3)2 + … + (5-3)2 + (6-3)2]/10}
=Sqrt{30/10}=1.73
b) In a month’s time (30 days), how many “laceration” packages do you expect
you’ll need? What about in two months (60 days)? Four months (120
days)?(Your answer should be of the form EV “give or take” SE )
Ans: EV(Sum)=(# draws from the box)*AVG(box)=(# days)*AVG(box)
So for 30 days ,EV(30)=(30)*3= 90
SE(Sum)=Sqrt (#draws from the box)*SD(box)
=Sqrt(30)*1.73 = 9.47
So in 30 days, you expect to use 90 give or take 10 laceration packs.
Similarly,
in 60 days, you expect to use 180 give or take 14 laceration packs.
in 120 days, you expect to use 360 give or take 19 laceration packs.
c) The % your SE is of your EV is found by looking at the ratio SE/EV. Find this
ratio for what you found above for one, two and four months. Compare these
results. What do you notice?
Ans: 9.47/90 = 10.5%
13.4/180= 7.4%
19/360=
5.3%
So, as the # of days increases, which is the same as the # of draws from the box,
the SE is smaller relative to the EV. That is, the SE grows with the # of draws
slower than the EV, and hence the sum “strays” less from the EV, percentage
wise, as the # draws from the box grows.
2) Consider a true or false test with 100 questions where your score is 1pt for each
correct answer. Since you have studied, you have an 80% chance of getting the right
answer.
a) Make a box model for your “score gain” for each question. Find mean and SD of
the box.
Ans: The box looks like [1,1,1,1,0].
AVG(box)=[(1+1+1+1+0)/5]=.8
SD(box) =Sqrt{[4*(1-.8)2 + (0-.8)2]/5} = .4
b) What do you expect your score to be? Again your answer should be EV “give or
take” SE.
Ans: Here, the # of draws from the box is the # of question on the test = 100, so
EV(sum)= 100*.8 = 80
SE(sum)= Sqrt(100)*.4=4
So you expect a score of 80 give or take 4.
3) Using a normal approximation to the sums in 1) and 2),
a) What is the chance of using more than 100 laceration packs in a month?
Z100= (100-EV90)/SE90 = (100-90)/9.5=1.05, which corresponds to an area under the
normal curve to the right of 1-.853 =.147 or 14.7%
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b) What is the chance of getting less than 75 correct answers on the true and
false exam?
Z75=(75-EV100)/SE100= (75-80)/4 = -1.25, which corresponds to an area under the
normal curve to the left of 1-.894=.106 or 10.6%
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