Suppose a researcher is interested in estimating the true percentage of individu

Question

Suppose a researcher is interested in estimating the true percentage of individuals who do not pay taxes on time. If the researcher is interested in a margin of error of at most 3%, what is the sample size required in order to get that margin of error with 95% confidence if there is evidence that historically the percentage of people who do not pay taxes on time is 37%| How does the estimate change if we are unable to give an initial estimate for the percentage of individuals? What is the new sample size required?

Explanation / Answer

2.

a)

Note that      
      
n = z(alpha/2)^2 p (1 - p) / E^2      
      
where      
      
alpha/2 =    0.025  
       
      
Using a table/technology,      
      
z(alpha/2) =    1.959963985  
      
Also,      
      
E =    0.03  
p =    0.37  
      
Thus,      
      
n =    994.9378346  
      
Rounding up,      
      
n =    995   [ANSWER]

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b)

Note that      
      
n = z(alpha/2)^2 p (1 - p) / E^2      
      
where      
      
alpha/2 =    0.025  
As there is no previous estimate for p, we set p = 0.5.      
      
Using a table/technology,      
      
z(alpha/2) =    1.959963985  
      
Also,      
      
E =    0.03  
p =    0.5  
      
Thus,      
      
n =    1067.071895  
      
Rounding up,      
      
n =    1068   [ANSWER]

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