Shortly after September 11th 2001, a researcher wanted to determine if the propo

Question

Shortly after September 11th 2001, a researcher wanted to determine if the proportion of females that favored war with Iraq was significantly different from the proportion of males that favored war with Iraq. In a sample of 70 females, 28 favored war with Iraq. In a sample of 66 males, 36 favored war with Iraq.

A) Let pF represent the proportion of females that favor the war, pM represent the proportion of males that favor the war. What are the proper hypotheses?

a) H0: pF < pM versus Ha: pF = pM

   b) H0: pF = pM versus Ha: pF < pM    

c) H0: pF = pM versus Ha: pF pM

d) H0: pF = pM versus Ha: pF > pM

B) What is the test statistic? Compute the statistic using male statistics subtracted from female statistics. Give your answer to four decimal places.  

C) What is the P-value for the test? Give your answer to four decimal places.

D) Using a 0.01 level of significance, what conclusion should be reached?

a) The proportion of females that favor the war and the proportion of males that favor the war are significantly different because the P-value is less than 0.01.

b) The proportion of females that favor the war and the proportion of males that favor the war are not significantly different because the P-value is less than 0.01.     

c) The proportion of females that favor the war and the proportion of males that favor the war are significantly different because the P-value is greater than 0.01.

d) The proportion of females that favor the war and the proportion of males that favor the war are not significantly different because the P-value is greater than 0.01.

E) What is the lower endpoint of a 99% confidence interval for the difference between the proportion of females that favor the war and the proportion of males that favor the war? Give your answer to four decimal places.  

F) What is the upper endpoint of a 99% confidence interval for the difference between the proportion of females that favor the war and the proportion of males that favor the war? Give your answer to four decimal places.

Explanation / Answer

a)

Formulating the hypotheses          
Ho: p1 - p2   =   0  
Ha: p1 - p2   =/=   0   [ANSWER, C]

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b)

Here, we see that pdo =    0   , the hypothesized population proportion difference.  
          
Getting p1^ and p2^,          
          
p1^ = x1/n1 =    0.545454545      
p2 = x2/n2 =    0.4      
          
Also, the standard error of the difference is          
          
sd = sqrt[ p1 (1 - p1) / n1 + p2 (1 - p2) / n2] =    0.084765237      
          
Thus,          
          
z = [p1 - p2 - pdo]/sd =    1.715969309   [ANSWER]

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c)  

Also, the p value is, as this is two tailed,
          
P =    0.08616766   [ANSWER]

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d)  
          
OPTION D: d) The proportion of females that favor the war and the proportion of males that favor the war are not significantly different because the P-value is greater than 0.01. [ANSWER]

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e)

For the   99%   confidence level, then  
          
alpha/2 = (1 - confidence level)/2 =    0.005      
z(alpha/2) =    2.575829304      
Margin of error = z(alpha/2)*sd =    0.218340782      
lower bound = p1^ - p2^ - z(alpha/2) * sd =    -0.072886236   [ANSWER]

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f)

Also,
  
upper bound = p1^ - p2^ + z(alpha/2) * sd =    0.363795327   [ANSWER]

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