Question

Suppose that 57% of all college seniors have a job prior to graduation. If a random sample of 80 college seniors is taken, approximate the probability that more than 46 have a job prior to graduation. Use the normal approximation to the binomial with a correction for continuity.

Answer 1

Let X be the number of college seniors who have a job prior to graduation                  
p = probability of selecting a college senior having a job = 0.57                  
n = 80                  
X follows Binomial distribution with n = 80 and p = 0.57                  

Using Normal Approximation for Binomial, we know                      
X ~ Normal distribution with mean μ=np and standard deviation σ=sqrt(npq)                      
q = 1-p = 1-0.57 = 0.43                      
μ = np = 80*0.57 = 45.6                      
σ =sqrt(npq) = sqrt(80*0.57*0.43) = 4.4281                      
we get                      
μ = 45.6                      
σ = 4.4281                      

To find P(more than 46 have a job prior to graduation)                      
that is to find P(X > 46)                      
                      
With Continuity Correction                      
P(X > 46) = 1 - P(X ≤ 46)                      
   = 1 - P(X ≤ 46.5)       …adding 0.5 for continuity correction          
                      
Using Excel function NORM.DIST to find the probability, we get                      
P(X > 46) = 1 - NORM.DIST(46.5, 45.6, 4.4281, TRUE)                      
   = 1 - 0.5805                  
   = 0.4195                  
P(more than 46 have a job prior to graduation) =