For this problem carry at least four digits after the decima

For this problem, carry at least four digits after the decimal in your calculations. Answers may vary slightly due to rounding.

A random sample of medical files is used to estimate the proportion p of all people who have blood type B.

(a) If you have no preliminary estimate for p, how many medical files should you include in a random sample in order to be 85% sure that the point estimate p will be within a distance of 0.07 from p? (Round your answer up to the nearest whole number.)
medical files

(b) Answer part (a) if you use the preliminary estimate that about 11 out of 90 people have blood type B.
medical files

Solution

A)

Note that      
      
n = z(alpha/2)^2 p (1 - p) / E^2      
      
where      
      
alpha/2 =    0.075  
As there is no previous estimate for p, we set p = 0.5.      
      
Using a table/technology,      
      
z(alpha/2) =    1.439531471  
      
Also,      
      
E =    0.07  
p =    0.5  
      
Thus,      
      
n =    105.7270845  
      
Rounding up,      
      
n =    106   [ANSWER]

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b)

Note that      
      
n = z(alpha/2)^2 p (1 - p) / E^2      
      
where      
      
alpha/2 =    0.075  
       
      
Using a table/technology,      
      
z(alpha/2) =    1.439531471  
      
Also,      
      
E =    0.07  
p =    0.122222222  
      
Thus,      
      
n =    45.37127724  
      
Rounding up,      
      
n =    46   [ANSWER]