Question

A politician has commissioned a survey of blue-collar and white-collar workers in her constituency. The survey asks each if they intend to vote for her. The results are presented below:

Vote     |   Blue Collar White-Collar

    For      |       448          265

  Against |       552          235     

a. Is there evidence of a significant difference in the proportions of workers that support her between the two groups of workers? Test using a .05 level of significance.

b. Find beta for the test if the true proportion voting for her among blue collar workers was 15 % less than for white collar workers.

Answer 1

a)

Ho:   p1 - p2 =   0          
Ha:   p1 - p2 ╪   0          
                  
sample #1   ----->              
first sample size,     n1=   1000          
number of successes, sample 1 =     x1=   448          
proportion success of sample 1 , p̂1=   x1/n1=   0.4480          
                  
sample #2   ----->              
second sample size,     n2 =    500          
number of successes, sample 2 =     x2 =    265          
proportion success of sample 1 , p̂ 2=   x2/n2 =    0.5300          
                  
difference in sample proportions, p̂1 - p̂2 =     0.4480   -   0.5300   =   -0.0820
                  
pooled proportion , p =   (x1+x2)/(n1+n2)=   0.4753          
                  
std error ,SE =    =SQRT(p*(1-p)*(1/n1+ 1/n2)=   0.02735          
Z-statistic = (p̂1 - p̂2)/SE = (   -0.082   /   0.0274   ) =   -2.9979
                  
p-value =        0.0027   [excel formula =2*NORMSDIST(z)]      
decision :    p-value<α,Reject null hypothesis               
                  
Conclusion:   There is enough evidence of a significant difference in the proportions of workers that support her between the two groups of workers

b)

We will fail to reject the null (commit a Type II error) if we get a Z statistic between                       -1.960   and   1.960
these Z-critical value corresponds to some X critical values ( X critical), such that                              
-1.960   ≤(p^ - po)/σpo≤   1.960  
-1.960   *σpo + po≤ p^ ≤   1.960   *σpo + po
-1.96*0.02735 + 0 ≤ p^ ≤ 1.96*0.02735 + 0

-0.0536≤ p^ ≤ 0.0536

std error ,SE =SQRT(p̂1*(1-p̂1)/n1 + p̂2*(1-p̂2)/n2)=   0.02730

now, type II error is ,ß =        P(-0.0536< p^ < 0.0536)       =P( (0.1216-p) /σp < Z < (0.2784-p)/σp )              
       =P( (-0.0536-0.15)/0.0273 < (X-µ)/σ < (0.0536-0.15)/0.0273 )                                      
                                      
P (    -7.458   < Z <    -3.531   )                       
= P ( Z <    -3.531   ) - P ( Z <   -7.46   ) =    0.0002   -    0.0000   =    0.0002   (answer)