Question

Recall that small effects may be statistically significant if the samples are large. A study of small-business failures looked at 145 food-and-drink businesses. Of these, 101 were headed by men and 44 were headed by women. During a three-year period, 14 of the men's businesses and 7 of the women's businesses failed.

(a) Find the proportions of failures for businesses headed by men (sample 1) and businesses headed by women (sample 2). These sample proportions are quite close to each other.

p̂men	=
p̂women	=

Give the P-value for the z test of the hypothesis that the same proportion of women's and men's businesses fail. (Use the two-sided alternative.) The test is very far from being significant. (Round your test statistic to two decimal places and your P-value to four decimal places.)

z	=
P-value	=

(b) Now suppose that the same sample proportions came from a sample of 30 times as large. That is, 210 out of 1320 business headed by women and 420 out of 3030 businesses headed by men fail. Verify that the proportions of failures are exactly the same as in (a). Repeat the z test for the new data, and show that it is now more significant. (Round your test statistic to two decimal places and your P-value to four decimal places.)

z	=
P-value	=

(c) Give the 95% confidence intervals for the difference between the proportions of men's and women's businesses that fail from Part (a) and Part (b).

For part (a):

95% CI =   ,

For part (b):

95% CI =   ,

(d) What is the effect of larger samples on the confidence interval?

The larger samples make the margin of error (and thus the length of the confidence interval) larger.The larger samples make the difference (and thus the length of the confidence interval) larger.    The larger samples make the margin of error (and thus the length of the confidence interval) smaller.The larger samples make the difference (and thus the length of the confidence interval) smaller.

Answer 1

a)

sample #1   ----->  
first sample size,     n1=   101
number of successes, sample 1 =     x1=   14
men , proportion success of sample 1 , p̂1=   x1/n1=   0.1386
      
sample #2   ----->  
second sample size,     n2 =    44
number of successes, sample 2 =     x2 =    7
women , proportion success of sample 1 , p̂ 2=   x2/n2 =    0.1591

difference in sample proportions, p̂1 - p̂2 =     0.1386   -   0.1591   =   -0.0205
                  
pooled proportion , p =   (x1+x2)/(n1+n2)=   0.1448          
                  
std error ,SE =    =SQRT(p*(1-p)*(1/n1+ 1/n2)=   0.06357          
Z-statistic = (p̂1 - p̂2)/SE = (   -0.020   /   0.0636   ) =   -0.32
                  
z-critical value , Z* =        1.9600   [excel formula =NORMSINV(α/2)]      
p-value =        0.7474   [excel formula =2*NORMSDIST(z)]      

b)

sample #1   ----->              
first sample size,     n1=   3030          
number of successes, sample 1 =     x1=   420          
proportion success of sample 1 , p̂1=   x1/n1=   0.1386          
                  
sample #2   ----->              
second sample size,     n2 =    1320          
number of successes, sample 2 =     x2 =    210          
proportion success of sample 1 , p̂ 2=   x2/n2 =    0.1591          
                  
difference in sample proportions, p̂1 - p̂2 =     0.1386   -   0.1591   =   -0.0205
                  
pooled proportion , p =   (x1+x2)/(n1+n2)=   0.1448          
                  
std error ,SE =    =SQRT(p*(1-p)*(1/n1+ 1/n2)=   0.01161          
Z-statistic = (p̂1 - p̂2)/SE = (   -0.020   /   0.0116   ) =   -1.76
                  
z-critical value , Z* =        1.9600   [excel formula =NORMSINV(α/2)]      
p-value =        0.0777   [excel formula =2*NORMSDIST(z)]      

c)

for part a)

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) =     0.06498          
margin of error , E = Z*SE =    1.960   *   0.0650   =   0.12736
                  
confidence interval is                   
lower limit = (p̂1 - p̂2) - E =    -0.020   -   0.1274   =   -0.1478
upper limit = (p̂1 - p̂2) + E =    -0.020   +   0.1274   =   0.1069
                  
so, confidence interval is (   -0.1478   < p1 - p2 <   0.1069   )  
---------------------

for part b)

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) =     0.01186          
margin of error , E = Z*SE =    1.960   *   0.0119   =   0.02325
                  
confidence interval is                   
lower limit = (p̂1 - p̂2) - E =    -0.020   -   0.0233   =   -0.0437
upper limit = (p̂1 - p̂2) + E =    -0.020   +   0.0233   =   0.0028
                  
so, confidence interval is (   -0.0437   < p1 - p2 <   0.0028   )  

d)

   The larger samples make the margin of error (and thus the length of the confidence interval) smaller.