Question

A study investigated whether regular mammograms resulted in fewer deaths from breast cancer over a period of 18 years. Among 30 comma 767 women who never had​ mammograms, 194 died of breast​ cancer, while only 146 of 30 comma 291 who had undergone screening died of breast cancer. ​a) Do these results suggest that mammograms may be an effective screening tool to reduce breast cancer​ deaths? ​b) If your conclusion is​ incorrect, which type of error did you​ commit?

Answer 1

a)

Ho:   p1 - p2 =   0          
Ha:   p1 - p2 >   0          
                  
sample #1   ----->              
first sample size,     n1=   30767          
number of successes, sample 1 =     x1=   194          
proportion success of sample 1 , p̂1=   x1/n1=   0.0063          
                  
sample #2   ----->              
second sample size,     n2 =    30291          
number of successes, sample 2 =     x2 =    146          
proportion success of sample 1 , p̂ 2=   x2/n2 =    0.0048          
                  
difference in sample proportions, p̂1 - p̂2 =     0.0063   -   0.0048   =   0.0015
                  
pooled proportion , p =   (x1+x2)/(n1+n2)=   0.0056          
                  
std error ,SE =    =SQRT(p*(1-p)*(1/n1+ 1/n2)=   0.00060          
Z-statistic = (p̂1 - p̂2)/SE = (   0.001   /   0.0006   ) =   2.4664
                  
z-critical value , Z* =        1.6449   [excel function =NORMSINV(α)]      
p-value =        0.0068   [excel function =NORMSDIST(-z)]      
decision :    p-value<α,Reject null hypothesis               
                  
Conclusion:   There is enough evidence to conlcude that mammograms may be an effective screening tool to reduce breast cancer​ deaths

b)

Type I error will be committed