Question

Use a normal approximation to find the probability of the indicated number of voters. In this​ case, assume that 160 eligible voters aged​ 18-24 are randomly selected. Suppose a previous study showed that among eligible voters aged​ 18-24, 22% of them voted.

Probability that fewer than 39 voted

The probability that fewer than 39 of 160 eligible voters voted is _____.

Answer 1

Let X be the number of voters voted from the sample of 160 voters                          
p = probability of selecting a voter who voted = 0.22                   (Since 22% voted)      
n = 160                          
X follows Binomial distribution with n = 160 and p = 0.22                          
PDF of X is given by                          
                   
Normal Approximation                          
                          
X follows Binomial distribution with n = 160 and p = 0.22                          
Using Normal Approximation for Binomial, we know                          
X ~ Normal distribution with mean μ=np and standard deviation σ=sqrt(npq)                          
q = 1-p = 1-0.22 = 0.78                          
μ= np = 160*0.22 = 35.2                         
σ=sqrt(npq) = sqrt(160*0.22*0.78) = 5.2398                          
To find P(fewer than 39 out of 160 eligible voters voted)                          
that is to find P(X < 39)                          
                          
With Continuity Correction                          
P(X < 39) = P(X < 39.5)                          
Using Excel function NORM.DIST to find the probability, we get                          
P(X < 39.5) = NORM.DIST(39.5, 35.2, 5.2398, TRUE)                          
   = 0.7941                      
P(fewer than 39 out of 160 eligible voters voted) = 0.7941             

          
                          
Without Continuity Correction                          
P(X < 39)                           
Using Excel function NORM.DIST to find the probability, we get                          
P(X < 39) = NORM.DIST(39, 35.2, 5.2398, TRUE)                          
   = 0.7658                      
P(fewer than 39 out of 160 eligible voters voted) = 0.7658                          
                          

Note ; Problem is solved using continuity correction and without continuity correction. The student may use

one of the solutions as prescribed in class