A researcher conducts a study of white and black attitudes t

A researcher conducts a study of white and black attitudes towards police in her community.

The percentage of a random sample of white respondents (N = 250) who say they have a favorable attitude towards the police is 51%. The percentage of a random sample of black respondents (N = 300) who say they have a favorable attitude toward the police is 47%.

You are asked if there is a real difference between the percentage of whites and blacks who have a positive attitude toward the police in the larger population, or is this sample difference likely to have occurred by random chance or sampling error.

How do you respond? Explain your answer.

Construct a 95% confidence interval for the proportion of blacks in the population who have a favorable attitude towards the police.

Solution

We set the significance level at be alpha = 0.05.

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 =    0.51      
p2 = x2/n2 =    0.47      
          
Also, the standard error of the difference is          
          
sd = sqrt[ p1 (1 - p1) / n1 + p2 (1 - p2) / n2] =    0.04277772      
          
Thus,          
          
z = [p1 - p2 - pdo]/sd =    0.935066197      
          
As significance level =    0.05   , then the critical z is  
          
zcrit =    1.959963985      
          
Also, the p value is          
          
P =    0.349754186      
          
Thus comparing z and zcrit (or p and significance level), then we    FAIL TO REJECT THE NULL HYPOTHESIS.      

Thus, there is no significant evidence at 0.05 level that there is a real difference between the percentage of whites and blacks who have a positive attitude toward the police in the larger population. [conclusion]

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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] =    0.028815505          
              
Now, for the critical z,              
alpha/2 =   0.025          
Thus, z(alpha/2) =    1.959963985          
Thus,              
              
lower bound = p^ - z(alpha/2) * sp =   0.413522648          
upper bound = p^ + z(alpha/2) * sp =    0.526477352          
              
Thus, the confidence interval is              
              
(   0.413522648   ,   0.526477352   ) [ANSWER, confidence interval of blacks]