1 62 of the children age 7 born in a particular city receive

1. 62% of the children age 7 born in a particular city receive all the required childhood vaccines. If a sample of 500 children are selected, find the probability that exactly 290 received all the required childhood vaccinations. (Hint: use the normal approximation to the binomial. First check to see if the conditions are met).

Solution

Here,

n p = 500*0.62 = 310 > 10
n q = 500*(1-0.62) = 190 > 10 [CONDITIONS MET!]

Also,

standard deviation = sqrt(n p q) = sqrt(500*0.62*(1-0.62)) = 10.85357084

We first get the z score for the two values. As z = (x - u) / s, then as          
x1 = lower bound =    289.5      
x2 = upper bound =    290.5      
u = mean =    310      
          
s = standard deviation =    10.85357084      
          
Thus, the two z scores are          
          
z1 = lower z score = (x1 - u)/s =    -1.888779306      
z2 = upper z score = (x2 - u) / s =    -1.79664373      
          
Using table/technology, the left tailed areas between these z scores is          
          
P(z < z1) =    0.029460703      
P(z < z2) =    0.036196099      
          
Thus, the area between them, by subtracting these areas, is          
          
P(z1 < z < z2) =    0.006735395   [ANSWER]