Two different simple random samples are drawn from two diffe

Two different simple random samples are drawn from two different populations. The first sample consists of 20 people with 11 having a common attribute. The second sample consists of 2000 people with 1439 of them having the same common attribute. Compare the results from a hypothesis test of P_1 = p_2 (with a 0.05 significance level) and a 95% confidence interval estimate of p_1 -P_2. What is the conclusion based on the hypothesis test? The 95% confidence interval is what is the conclusion based on the confidence interval? How do the results from the hypothesis test and the confidence interval compare?

Solution

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 =    0.55      
p2 = x2/n2 =    0.7195      
          
Also, the standard error of the difference is          
          
sd = sqrt[ p1 (1 - p1) / n1 + p2 (1 - p2) / n2] =    0.111695613      
          
Thus,          
          
z = [p1 - p2 - pdo]/sd =    -1.517517081      
          
As significance level =    0.05   , then the critical z is  
          
zcrit =    1.959963985      
          
Also, the p value is          
          
P =    0.129136184      
          
Thus,
*******************************

The test statistic is OUTSIDE the critical region, so WE FAIL TO REJECT the null hypothesis. There is NO SIGNIFICANT evidence to conclude that p1=/=p2. [ANSWER]

*******************************

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 =    -0.388419378      
upper bound = p1^ - p2^ + z(alpha/2) * sd =    0.049419378      
          
Thus, the confidence interval is          
          
(   -0.388419378   ,   0.049419378 ) [ANSWER, CONFIDENCE INTERVAL]

************************************

Since 0 is INCLUDED in the interval, it indicates to FAIL TO REJECT the null hypothesis. [ANSWER]

************************************

The results are THE SAME, since the hypothesis tests suggests that p1=p2, and the confidence interval suggests that p1 = p2. [ANSWER]