Question

Construct a confidence interval for p 1 minus p 2 p1−p2 at the given level of confidence. x 1 equals x1= 375 375​, n 1 equals n1= 523 523​, x 2 equals x2= 432 432​, n 2 equals n2= 585 585​, 95 95​% confidence The researchers are nothing ​% confident the difference between the two population​ proportions, p 1 minus p 2 p1−p2​, is between nothing and nothing . ​(Use ascending order. Type an integer or decimal rounded to three decimal places as​ needed.) Thank you

Answer 1

sample #1   ----->  
first sample size,     n1=   523
number of successes, sample 1 =     x1=   375
proportion success of sample 1 , p̂1=   x1/n1=   0.7170
      
sample #2   ----->  
second sample size,     n2 =    585
number of successes, sample 2 =     x2 =    432
proportion success of sample 1 , p̂ 2=   x2/n2 =    0.7385

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) =     0.0268          
margin of error , E = Z*SE =    1.960   *   0.0268   =   0.0525
                  
confidence interval is                   
lower limit = (p̂1 - p̂2) - E =    -0.021   -   0.0525   =   -0.0740
upper limit = (p̂1 - p̂2) + E =    -0.021   +   0.0525   =   0.0311
                  
so, confidence interval is (   -0.074 < p1 - p2 <   0.031 )