The financial department at a large hospital would like to estimate the average

Question

The financial department at a large hospital would like to estimate the average outstanding balance owed by patients who have not paid their bills in full. In order for the interval to be useful in making budgeting decisions the department's management staff requires the the interval be no wider than $700 (total) at a confidence level of 99%. Ten files of patients carrying outstanding balances were randomly selected from a list of all of all pediatrics patients and the outstanding balance (rounded to the nearest integer) was recorded. The data are reported below in Table 1. This can be considered a pilot study.  

A) The total number of patient records that must be sampled in order for the finance dept. management to meet their constraints of confidence level and bound is:

Table 1: Outstanding balances ($) for 10 randomly selected patients 8343 4677 7469 6337 7136 8947 4719 8741 7607 3913

Explanation / Answer

Getting the mean, X,          
          
X = Sum(x) / n          
Sum(x) =    67889      
As n =    10      
Thus,          
X =    6788.9      
          
Setting up tables,          
x   x - X   (x - X)^2  
8343   1554.1   2415226.81  
4677   -2111.9   4460121.61  
7469   680.1   462536.01  
6337   -451.9   204213.61  
7136   347.1   120478.41  
8947   2158.1   4657395.61  
4719   -2069.9   4284486.01  
8741   1952.1   3810694.41  
7607   818.1   669287.61  
3913   -2875.9   8270800.81  
          
          
          
Thus, Sum(x - X)^2 =    29355240.9      
          
Thus, as           
          
s^2 = Sum(x - X)^2 / (n - 1)          
          
As n =    10      
          
s^2 =    3261693.433      
          
Thus,          
          
s =    1806.015901   [STANDARD DEVIATION]

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Note that      
      
n = z(alpha/2)^2 s^2 / E^2      
      
where      
      
alpha/2 = (1 - confidence level)/2 =    0.005  
      
Using a table/technology,      
      
z(alpha/2) =    2.575829304  
      
Also,      
      
s = sample standard deviation =    1806.015901  
E = margin of error = width/2 = 700/2 = 350  
      
Thus,      
      
n =    176.6612137  
      
Rounding up,      
      
n =    177   [ANSWER]

  

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