A study was recently published on the relationship between m

A study was recently published on the relationship between method of conception and prevalence of major birth defects. The prevalence of at least one major birth defect among infants conceived naturally was 4.2%, based on a large sample of infants. Among 837 infants born as a result of IVF, 75 had at least one major birth defect.

Do an unusual number of infants have at least one birth defect in the IVF group? Why or why not? (Use an approximation to the binomial distribution to calculate the probability of getting no less than 75 non-normal infants.)

Solution

Here,

n = 837

p = 0.042

Thus,

mean = n p = 837*0.042 = 35.154

standard deviation = sqrt(np(1-p)) = sqrt(837*0.042*(1-0.042)) = 5.803234615

We first get the z score for the critical value. As z = (x - u) / s, then as          
          
x = critical value =    74.5      
u = mean =    35.154      
          
s = standard deviation =    5.803234615      
          
Thus,          
          
z = (x - u) / s =    6.780011943      
          
Thus, using a table/technology, the right tailed area of this is          
          
P(z >   6.780011943   ) =    6.0083*10^-12 or 0.0000000000060083

As this is a very small probability, then yes, this is UNUSUAL.