Question

Never forget that even small effects can be statistically significant if the samples are large. To illustrate this fact, consider a sample of 144 small businesses. During a three-year period, 14 of the 99 headed by men and 8 of the 45 headed by women failed.

(a) Find the proportions of failures for businesses headed by women and businesses headed by men. These sample proportions are quite close to each other. Give the P-value for the test of the hypothesis that the same proportion of women's and men's businesses fail. (Use the two-sided alternative). What can we conclude (Use α=0.05α=0.05)?
The P-value was  so we conclude that
Choose a conclusion. The test showed strong evidence of a significant difference. The test showed no significant difference.

(b) Now suppose that the same sample proportion came from a sample 30 times as large. That is, 240 out of 1350 businesses headed by women and 420 out of 2970 businesses headed by men fail. Verify that the proportions of failures are exactly the same as in part (a). Repeat the test for the new data. What can we conclude?
The P-value was  so we conclude that
Choose a conclusion. The test showed strong evidence of a significant difference. The test showed no significant difference.

(c) It is wise to use a confidence interval to estimate the size of an effect rather than just giving a P-value. Give 95% confidence intervals for the difference between proportions of men's and women's businesses (men minus women) that fail for the settings of both (a) and (b). (Be sure to check that the conditions are met. If the conditions aren't met for one of the intervals, use the same type of interval for both)
Interval for smaller samples:  to  
Interval for larger samples:

Answer 1

a)

Ho:   p1 - p2 =   0          
Ha:   p1 - p2 ╪   0          
                  
sample #1   ----->              
first sample size,     n1=   99          
number of successes, sample 1 =     x1=   14          
proportion success of sample 1 , p̂1=   x1/n1=   0.1414141          
                  
sample #2   ----->              
second sample size,     n2 =    45          
number of successes, sample 2 =     x2 =    8          
proportion success of sample 1 , p̂ 2=   x2/n2 =    0.177778          
                  
difference in sample proportions, p̂1 - p̂2 =     0.1414   -   0.1778   =   -0.0364
                  
pooled proportion , p =   (x1+x2)/(n1+n2)=   0.1527778          
                  
std error ,SE =    =SQRT(p*(1-p)*(1/n1+ 1/n2)=   0.06468          
Z-statistic = (p̂1 - p̂2)/SE = (   -0.036   /   0.0647   ) =   -0.562
                  

p-value =        0.5740   [excel formula =2*NORMSDIST(z)]      

The test showed no significant difference.

b)

Ho:   p1 - p2 =   0          
Ha:   p1 - p2 ╪   0          
                  
sample #1   ----->              
first sample size,     n1=   1350          
number of successes, sample 1 =     x1=   240          
proportion success of sample 1 , p̂1=   x1/n1=   0.1777778          
                  
sample #2   ----->              
second sample size,     n2 =    2970          
number of successes, sample 2 =     x2 =    420          
proportion success of sample 1 , p̂ 2=   x2/n2 =    0.141414          
                  
difference in sample proportions, p̂1 - p̂2 =     0.1778   -   0.1414   =   0.0364
                  
pooled proportion , p =   (x1+x2)/(n1+n2)=   0.1527778          
                  
std error ,SE =    =SQRT(p*(1-p)*(1/n1+ 1/n2)=   0.01181          
Z-statistic = (p̂1 - p̂2)/SE = (   0.036   /   0.0118   ) =   3.079
                  
  
p-value =        0.0021   [excel formula =2*NORMSDIST(z)]      
.The test showed strong evidence of a significant difference.

c)

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) =     0.06689          
margin of error , E = Z*SE =    1.960   *   0.0669   =   0.13111
                  
confidence interval is                   
lower limit = (p̂1 - p̂2) - E =    -0.036   -   0.1311   =   -0.1674721
upper limit = (p̂1 - p̂2) + E =    -0.036   +   0.1311   =   0.0947448
                  
so, confidence interval is (   -0.1675   < p1 - p2 <   0.0947   )  

--------------------

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) =     0.01221          
margin of error , E = Z*SE =    1.960   *   0.0122   =   0.02394
                  
confidence interval is                   
lower limit = (p̂1 - p̂2) - E =    -0.036   -   0.0239   =   -0.0603007
upper limit = (p̂1 - p̂2) + E =    -0.036   +   0.0239   =   -0.0124266
                  
so, confidence interval is (   -0.0603   < p1 - p2 <   -0.0124   )