In the 2004 presidential election exit polls from the critic

In the 2004 presidential election, exit polls from the critical state of Ohio provided the following results: For respondents with college degrees, 53% voted for Bush and 46% voted for Kerry. There were 2020 respondents. (a) Is there a significant difference in these proportions? Use = 0.05. What is the Pvalue? (b) Calculate a 95% confidence interval for the difference in the two proportions and comment on the use of this interval to answer the question in part (a).

Solution

a)

Formulating the hypotheses          
Ho: p1 - p2   =   0  
Ha: p1 - p2   =/=   0  
Here, we see that pdo =    0   , the hypothesized population proportion difference.  
          
Getting p1^ and p2^,          
          
p1^ = x1/n1 =    0.53      
p2 = x2/n2 =    0.46      
          
Also, the standard error of the difference is          
          
sd = sqrt[ p1 (1 - p1) / n1 + p2 (1 - p2) / n2] =    0.015693538      
          
Thus,          
          
z = [p1 - p2 - pdo]/sd =    4.460434663      
          
As significance level =    0.05   , then the critical z is  
          
zcrit =    1.959963985      
          
Also, the p value is          
          
P =    8.17936*10^-6      
          
As z>1.96, and P < 0.05, then we    REJECT THE NULL HYPOTHESIS.      

Thus, there is significant evidence that there is a difference in these proportions.

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b)

For the   95%   confidence level, then  
          
alpha/2 = (1 - confidence level)/2 =    0.025      
z(alpha/2) =    1.959963985      
          
lower bound = p1^ - p2^ - z(alpha/2) * sd =    0.039241231      
upper bound = p1^ - p2^ + z(alpha/2) * sd =    0.100758769      
          
Thus, the confidence interval is          
          
(   0.039241231   ,   0.100758769 )

As this whole interval is greater than 0, then we confirm part a) that there is significant difference between the proportions.