Umbrella Corporation is trying to determine whether a new dr

Umbrella Corporation is trying to determine whether a new drug improves mood in individuals with bipolar disorder. The company gave the drug to 253 individuals (Group 1) and a placebo (or sugar pill) to a separate sample of 238 individuals (Group 2). After one month on the drug, 166 of the patients in Group 1 reported improved mood, while 197 of the patients in Group 2 reported the same.

Calculate the lower bound of a 95% confidence interval for the true difference in the proportion of individuals who experienced improved mood, to two decimal places

Solution

Here, we see that pdo =    0   , the hypothesized population proportion difference.  
          
Getting p1^ and p2^,          
          
p1^ = x1/n1 =    0.656126482      
p2 = x2/n2 =    0.827731092      
          
Also, the standard error of the difference is          
          
sd = sqrt[ p1 (1 - p1) / n1 + p2 (1 - p2) / n2] =    0.038612485      
          
Thus,          
          
          
For the   90%   confidence level, then  
          
alpha = (1 - confidence level) =    0.05      

z(alpha) =    1.644853627      

Thus,
          
lower bound = p1^ - p2^ - z(alpha/2) * sd = -0.235116496 = -0.24 [ANSWER]