What is the probability of observing a value X of 25 or less

Question

1. Data collected over a long period of time showed that a particular genetic defect occurs in 2 of every 1000 children. Let X be the random variable “number of children with genetic defect in a sample of 20,000 children examined”. The records of a medical clinic show X = 25 children.

a) What is the probability of observing a value X of 25 or less?

b) Would you say that the observation of X = 25 children with genetic defect was rather unlikely?

please help it will be on test.

Answer 1

2 in 1000 gives a mean of 0.2%=0.002. In 20000 children µ, the mean, is 0.002*20000=40. So we would expect 40 cases in a sample size of 20,000.

a) Assuming a binomial distribution, in the absence of any other statistic, SD=√(20000*0.002*0.998)=6.32 approx. If we use this as σ in a normal distribution the z-score for X=25 is (25-40)/6.32=-2.37 corresponding to 0.0089 or 0.89%.

b) It is unlikely for X=25 given the mean and SD calculated.