500 voters were randomly interviewed, and it was found 235 out of the 500 would

Question

500 voters were randomly interviewed, and it was found 235 out of the 500 would vote for canidate X. Based on the information, construct a 95% confidence interval for the proportion of likely voters who will vote for canidate X. A) Verify any requirements. B) Confidence Interval C) Interpret this interval D) What is the margin error (MOE)? E) If the researchers wanted to perform another similar study with 95% confidence, what should they do in order to decrease the margin of error? F) Another report with 95% confidence is done and they want to lower the margin of error and have an MOE with 0.03. Determine the sample size n necessary to have a MOE within 0.03, using 0.47 as the point estimate. 500 voters were randomly interviewed, and it was found 235 out of the 500 would vote for canidate X. Based on the information, construct a 95% confidence interval for the proportion of likely voters who will vote for canidate X. A) Verify any requirements. B) Confidence Interval C) Interpret this interval D) What is the margin error (MOE)? E) If the researchers wanted to perform another similar study with 95% confidence, what should they do in order to decrease the margin of error? F) Another report with 95% confidence is done and they want to lower the margin of error and have an MOE with 0.03. Determine the sample size n necessary to have a MOE within 0.03, using 0.47 as the point estimate. A) Verify any requirements. B) Confidence Interval C) Interpret this interval D) What is the margin error (MOE)? E) If the researchers wanted to perform another similar study with 95% confidence, what should they do in order to decrease the margin of error? F) Another report with 95% confidence is done and they want to lower the margin of error and have an MOE with 0.03. Determine the sample size n necessary to have a MOE within 0.03, using 0.47 as the point estimate.

Explanation / Answer

A) Verify any requirements.

As the number of successes (235) and failures (265) are both greater than 10, then we can use normal approximation.

B) Confidence Interval

Note that              
              
p^ = point estimate of the population proportion = x / n =    0.47          
              
Also, we get the standard error of p, sp:              
              
sp = sqrt[p^ (1 - p^) / n] =    0.022320394          
              
Now, for the critical z,              
alpha/2 =   0.025          
Thus, z(alpha/2) =    1.959963985          
Thus,              
Margin of error = z(alpha/2)*sp =    0.043747169          
lower bound = p^ - z(alpha/2) * sp =   0.426252831          
upper bound = p^ + z(alpha/2) * sp =    0.513747169          
              
Thus, the confidence interval is              
              
(   0.426252831   ,   0.513747169   ) [ANSWER]

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C) Interpret this interval

We are 95% confident that the true population proportion of those who will vote for candidate X is betwee 0.4263 and 0.5137.

D) What is the margin error (MOE)?

Thus,              

Margin of error = z(alpha/2)*sp =    0.043747169   [ANSWER]

E) If the researchers wanted to perform another similar study with 95% confidence, what should they do in order to decrease the margin of error?

They can increase the sample size.

F) Another report with 95% confidence is done and they want to lower the margin of error and have an MOE with 0.03. Determine the sample size n necessary to have a MOE within 0.03, using 0.47 as the point estimate.

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 =    0.47  
      
Thus,      
      
n =    1063.230436  
      
Rounding up,      
      
n =    1064   [ANSWER]

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