Question

Two random samples are taken, one from among UVA students and the other from among UNC students. Both groups are asked if academics are their top priority. A summary of the sample sizes and proportions of each group answering yes'' are given below:

UVA (Pop. 1):n1=80, p̂ 1=0.72

UNC (Pop. 2):n2=81, p̂ 2=0.64

Find a 97.7% confidence interval for the difference p1−p2 of the population proportions.

Confidence interval = ?????

-any chance you can also explain how this is done on a calculator? If not possible, I understand. Please help me solve it.

Answer 1

sample #1   ----->  
first sample size,     n1=   80

proportion success of sample 1 , p̂1=    0.7200000
      
sample #2   ----->  
second sample size,     n2 =    81

proportion success of sample 1 , p̂ 2=      0.640000

α=1-0.977=0.023

Z critical value =   Z α/2 =    2.273   [excel function: =normsinv(α/2)      
                  
Std error , SE =    SQRT(p̂1 * (1 - p̂1)/n1 + p̂2 * (1-p̂2)/n2) =     0.07324          
margin of error , E = Z*SE =    2.273   *   0.0732   =   0.16651
                  
confidence interval is                   
lower limit = (p̂1 - p̂2) - E =    0.080   -   0.1665   =   -0.0865117
upper limit = (p̂1 - p̂2) + E =    0.080   +   0.1665   =   0.2465117
                  
so, confidence interval is (   -0.0865   < p1 - p2 <   0.2465   )  

=================

number of successes, sample 1 =     x1= np^ = 80*0.72=57.6

number of successes, sample 2 =     x2= 81*0.64 = 51.84

calculator steps-

press the STAT, and highlights TESTS

2-PropZInt(x1,n1,x2,n2,CL)
2-PropZInt(57.6,80,51.84,.95)