Among 2300 randomly selected male car occupants over the age of 8, 82% wear seat

Question

Among 2300 randomly selected male car occupants over the age of 8, 82% wear seatbelts. Among 2600 randomly selected female car occupants over the age of 8, 72% wear seat belts. Use a 0.1 significance level to test the claim that both genders have the same rate of seat belt use. Does there appear to be a gender gap? Does there appear to be a gender gap? A. There does not appear to be a gender gap because there is not a significant difference in the proportions. B. There appears to be a gender gap because there is not a significant difference in the proportions. C. There appears to be a gender gap because there is a significant difference in the proportions. D. There does not appear to be a gender gap because there is a significant difference in the proportions.

Explanation / Answer

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 =    0.82      
p2 = x2/n2 =    0.72      
          
Also, the standard error of the difference is          
          
sd = sqrt[ p1 (1 - p1) / n1 + p2 (1 - p2) / n2] =    0.011904301      
          
Thus,          
          
z = [p1 - p2 - pdo]/sd =    8.400325456      
          
As significance level =    0.1   , then the critical z is  
          
zcrit =    1.644853627      
          
Also, the p value is          
          
P =    4.45243E-17      
          
As |z| > 1.644, and P < 0.1 then we    REJECT THE NULL HYPOTHESIS.      

Hence,

OPTION C: There appears to be a gender gap because there is a significant difference in the proportions. [ANSWER]

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