28 and 29 refer to the following setup: In a random sample of 100 homes in a cer

Question

28 and 29 refer to the following setup: In a random sample of 100 homes in a certain city, it is found that 8 are heated by oil. What is the lower limit of a 95% confidence interval for the proportion of homes in that city heated by oil? Answer: 29. Investigators plan to sample homes in another city to estimate the proportion of those heated by oil. We have no prior knowledge about the proportion of homes in that city that might be heated by oil. How many homes should they sample to obtain a 95% confidence interval for the true city proportion heated with oil that has width at most 0.05?

Explanation / Answer

POPULATION PROPORTION (without p)              
Note that              
              
p^ = point estimate of the population proportion = x / n =    0.08          
              
Also, we get the standard error of p, sp:              
              
sp = sqrt[p^ (1 - p^) / n] =    0.02712932          
              
Now, for the critical z,              
alpha/2 =   0.025          
Thus, z(alpha/2) =    1.959963985          

Thus,              
              
lower bound = p^ - z(alpha/2) * sp =    0.02682751 [ANSWER]
          
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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 = width/2 = 0.025  
p =    0.5  
      
Thus,      
      
n =    1536.583528  
      
Rounding up,      
      
n =    1537   [ANSWER]

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