Question

1. You and a group of friends wish to start a company. You have an idea, and you are comparing startup incubators to apply to. (Start up incubators hold classes and help startups to contact venture capitalists and network with one another) Assume funding is normally distributed.

                     Incubator A has a 70% success ratio getting companies to survive at least 4 years from inception. The average venture funding of the 28 companies reaching that 4 year mark, is 1.3 million dollars with a standard deviation of 0.6 million

Incubator B has a 40% success ratio getting companies to survive at least 4 years from inception. The average venture funding of the 20 companies reaching that 4 year mark, is 1.9 million dollars with a standard deviation of 0.55 million

1. Are the success ratios significantly different?(note the count is given but not N, how do you find N?)

2. Is the average funding in incubator B significantly different from the average funding in a. (use a=0.01). Assume a normal distribution

Answer 1

Please note that alpha for part has been taken =0.01

A)

Ho:   p1 - p2 =   0          
Ha:   p1 - p2 ╪   0          
                  
sample #1   ----->   A          
first sample size,     n1=   28          

proportion success of sample 1 , p̂1=   x1/n1=   0.7000          
                  
sample #2   ----->   B          
second sample size,     n2 =    20          

proportion success of sample 1 , p̂ 2=   x2/n2 =    0.400          
                  
difference in sample proportions, p̂1 - p̂2 =     0.7000   -   0.4000   =   0.3000
                  
pooled proportion , p =   (x1+x2)/(n1+n2)=   0.5750          
                  
std error ,SE =    =SQRT(p*(1-p)*(1/n1+ 1/n2)=   0.1447          
Z-statistic = (p̂1 - p̂2)/SE = (   0.300   /   0.1447   ) =   2.0728
                  

p-value =        0.0382   [excel formula =2*NORMSDIST(z)]      
p-value>α,Don't reject null hypothesis   
              
                  
Conclusion:   There is enough evidence to support that the success ratios significantly do not different

b)

Ho :   µ1 - µ2 =   0                  
Ha :   µ1-µ2 ╪   0                  
                          
Level of Significance ,    α =    0.01                  
                          
Sample #1   ---->   A               
mean of sample 1,    x̅1=   1.30                  
standard deviation of sample 1,   s1 =    0.60                  
size of sample 1,    n1=   28                  
                          
Sample #2   ---->   B                
mean of sample 2,    x̅2=   1.90                  
standard deviation of sample 2,   s2 =    0.55                  
size of sample 2,    n2=   20                  
                          
difference in sample means =    x̅1-x̅2 =    1.3000   -   1.9   =   -0.60  
                          
pooled std dev , Sp=   √([(n1 - 1)s1² + (n2 - 1)s2²]/(n1+n2-2)) =    0.5799                  
std error , SE =    Sp*√(1/n1+1/n2) =    0.1698                  
                          
t-statistic = ((x̅1-x̅2)-µd)/SE = (   -0.6000   -   0   ) /    0.17   =   -3.534
                          
Degree of freedom, DF=   n1+n2-2 =    46                  
t-critical value , t* =        2.6870   (excel formula =t.inv(α/2,df)              
Decision:   | t-stat | > | critical value |, so, Reject Ho                      
p-value =        0.000944   (excel function: =T.DIST.2T(t stat,df) )              
Conclusion:     p-value <α , Reject null hypothesis      

               

  Conclusion:   There is enough evidence to support that the average funding in incubator B significantly different from the average funding in A

Please revert back in case of any doubt.

Please upvote. Thanks in advance.