To estimate the proportion of Democrats in a large city a ra

To estimate the proportion of Democrats in a large city, a random sample of 500 voters was taken: 260 were found to be Democrats.

Determine a 95% confidence Interval for the true proportion of Democrats:

Can we be 95% confident that at least half the city workers are Democrats?

if we had wanted a 95%confidence interval that was no wider than 0.05, how big should our sample size be?

Solution

a)

Note that              
              
p^ = point estimate of the population proportion = x / n =    0.52          
              
Also, we get the standard error of p, sp:              
              
sp = sqrt[p^ (1 - p^) / n] =    0.022342784          
              
Now, for the critical z,              
alpha/2 =   0.025          
Thus, z(alpha/2) =    1.959963985          
Thus,              
              
lower bound = p^ - z(alpha/2) * sp =   0.476208948          
upper bound = p^ + z(alpha/2) * sp =    0.563791052          
              
Thus, the confidence interval is              
              
(   0.476208948   ,   0.563791052   ) [ANSWER]

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b)

No, because not the whole interval is above 0.50.

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c)

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 =    0.025  
p =    0.52  
      
Thus,      
      
n =    1534.124995  
      
Rounding up,      
      
n =    1535   [ANSWER]