Question

Construct a confidence interval for p1−p2 at the given level of confidence. x1= 396​, n1= 503​, x2= 424​, n2= 587​, 95​% confidence

The researchers are __?__​% confident the difference between the two population​ proportions,p1−p2​, is between __?__and __?__.

​(Use ascending order. Type an integer or decimal rounded to three decimal places as​ needed.)

Answer 1

sample #1   ----->   sample 1
first sample size,     n1=   503
number of successes, sample 1 =     x1=   396
proportion success of sample 1 , p̂1=   x1/n1=   0.7873
      
sample #2   ----->   sample 2
second sample size,     n2 =    587
number of successes, sample 2 =     x2 =    424
proportion success of sample 1 , p̂ 2=   x2/n2 =    0.7223

level of significance, α =   0.05              
Z critical value =   Z α/2 =    1.960   [excel function: =normsinv(α/2)      
                  
Std error , SE =    SQRT(p̂1 * (1 - p̂1)/n1 + p̂2 * (1-p̂2)/n2) =     0.0260          
margin of error , E = Z*SE =    1.960   *   0.0260   =   0.0509
                  
confidence interval is                   
lower limit = (p̂1 - p̂2) - E =    0.065   -   0.0509   =   0.0141
upper limit = (p̂1 - p̂2) + E =    0.065   +   0.0509   =   0.1159
                  
so, confidence interval is (   0.0141   < p1 - p2 <   0.1159   )  

The researchers are 95% ​% confident the difference between the two population​ proportions,p1−p2​, is between 0.014 and 0.116.