Question

Topic: Comparing support for including ethnic consideration in college admissions between White and Black Populations.

Data: Whites (659 of 2,997 say Yes, race should be a factor) Blacks (577 of 1,518 say Yes, race should be a factor)

Provide the following:

1. Summary Statistics

2. T-Test (or Confidence Interval)

3. Hypothesis Test

4. Regression

Answer 1

1)

sample #1   ----->  
first sample size,     n1=   2997
number of successes, sample 1 =     x1=   659
proportion success of sample 1 , p̂1=   x1/n1=   0.2199
      
sample #2   ----->  
second sample size,     n2 =    1518
number of successes, sample 2 =     x2 =    577
proportion success of sample 1 , p̂ 2=   x2/n2 =    0.3801

2) α=0.05

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) =     0.01458          
margin of error , E = Z*SE =    1.960   *   0.0146   =   0.02857
                  
confidence interval is                   
lower limit = (p̂1 - p̂2) - E =    -0.160   -   0.0286   =   -0.1888
upper limit = (p̂1 - p̂2) + E =    -0.160   +   0.0286   =   -0.1317
                  
so, confidence interval is (   -0.1888   < p1 - p2 <   -0.1317   )  

3)

Ho:   p1 - p2 =   0
Ha:   p1 - p2 ╪   0

difference in sample proportions, p̂1 - p̂2 =     0.2199   -   0.3801   =   -0.1602
                  
pooled proportion , p =   (x1+x2)/(n1+n2)=   0.2738          
                  
std error ,SE =    =SQRT(p*(1-p)*(1/n1+ 1/n2)=   0.01405          
Z-statistic = (p̂1 - p̂2)/SE = (   -0.160   /   0.0140   ) =   -11.4062
                  

p-value =        0.0000   [excel formula =2*NORMSDIST(z)]      
decision :    p-value<α,Reject null hypothesis