Question

In a 4-week study about the effectiveness of using magnetic insoles to treat plantar heel pain, 17 of 54 subjects who wore magnetic insoles felt all or mostly better, while 18 of 41 subjects who wore non- magnetic insoles felt all or mostly better.

a) At 4% significance level, can we support a claim that there is a difference in the proportion of subjects who feel all or mostly better between the two groups? (Use the critical value approach and the P-value approach).

b) Find and interpret a 97% confidence interval for the difference in proportion of subjects who feel better in the two groups.

Answer 1

a)

Ho:   p1 - p2 =   0  
Ha:   p1 - p2 ╪   0  
          
first sample size,     n1=   54  
number of successes, sample 1 =     x1=   17  
proportion success of sample 1 , p̂1=   x1/n1=   0.315  
          
second sample size,     n2 =    41  
number of successes, sample 2 =     x2 =    18  
proportion success of sample 1 , p̂ 2=   x2/n2 =    0.439  
          
difference in sample proportions,    p̂1 - p̂2 =     -0.1242   
          
pooled proportion , p =   (x1+x2)/(n1+n2)=   0.3684  
          
std error ,SE =    =SQRT(p*(1-p)*(1/n1+ 1/n2)=   0.0999  
          
Z-statistic =    (p̂1 - p̂2)/SE = (-0.1242)/0.0999 = -1.2431  
          

critical value approach

z-critical value , Z* =        2.0537   [excel formula =NORMSINV(α/2)]

z stat <critical value , fail to reject Ho

p-value =        0.2138   [excel formula =2*NORMSDIST(z)]
decision :    p-value>α,Don't reject null hypothesis

there is not enough evidence to support a claim that there is a difference in the proportion of subjects who feel all or mostly better between the two groups at α=0.04

b)

α=0.03

Z critical value =   Z α/2 =    2.1701   [excel function: =normsinv(α/2)

Std error , SE =    SQRT(p̂1 * (1 - p̂1)/n1 + p̂2 * (1-p̂2)/n2) =     0.10

margin of error , E =   Z*SE = 2.1701*0.10 = 0.2170      
              
confidence interval is               
lower limit =    (p̂1 - p̂2) - E = -0.124 -0.2170 = -0.3412      
upper limit =   (p̂1 - p̂2) + E = -0.124 + 0.2170 = 0.0928      
              
so, confidence interval is (   -0.3412   < p1 - p2 <   0.0928   )