1In an article about the cost of health care Money magazine

1.In an article about the cost of health care, Money magazine reported that a visit to a hospital emergency room for something as simple as a sore throat has a mean cost of $328 (Money, January 2009). Assume that the cost for this type of hospital emergency room visit is normally distributed with a standard deviation of $92. Answer the following questions about the cost of a hospital emergency room visit for this medical service.

a.What is the probability that the cost will be more than $500?

b.What is the probability that the cost will be less than $250?

c.What is the probability that the cost will be between $300 and $400?

d.If the cost to a patient is in the lower 8% of charges for this medical service, what was the cost of this patient’s emergency room visit?

Solution

A)

We first get the z score for the critical value. As z = (x - u) / s, then as          
          
x = critical value =    500      
u = mean =    328      
          
s = standard deviation =    92      
          
Thus,          
          
z = (x - u) / s =    1.869565217      
          
Thus, using a table/technology, the right tailed area of this is          
          
P(z >   1.869565217   ) =    0.03077211 [answer]

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B)

We first get the z score for the critical value. As z = (x - u) / s, then as          
          
x = critical value =    250      
u = mean =    328      
          
s = standard deviation =    92      
          
Thus,          
          
z = (x - u) / s =    -0.847826087      
          
Thus, using a table/technology, the left tailed area of this is          
          
P(z <   -0.847826087   ) =    0.198267416 [ANSWER]

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c)

We first get the z score for the two values. As z = (x - u) / s, then as          
x1 = lower bound =    300      
x2 = upper bound =    400      
u = mean =    328      
          
s = standard deviation =    92      
          
Thus, the two z scores are          
          
z1 = lower z score = (x1 - u)/s =    -0.304347826      
z2 = upper z score = (x2 - u) / s =    0.782608696      
          
Using table/technology, the left tailed areas between these z scores is          
          
P(z < z1) =    0.380431456      
P(z < z2) =    0.783071533      
          
Thus, the area between them, by subtracting these areas, is          
          
P(z1 < z < z2) =    0.402640077   [ANSWER]

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d)

First, we get the z score from the given left tailed area. As          
          
Left tailed area =    0.08      
          
Then, using table or technology,          
          
z =    -1.40507156      
          
As x = u + z * s,          
          
where          
          
u = mean =    328      
z = the critical z score =    -1.40507156      
s = standard deviation =    92      
          
Then          
          
x = critical value =    198.7334165   [ANSWER]