sampling distribution of a proportion

When in a population we proceed to study a characteristic with only two possible values (success/failure), then the population follows a binomial distribution.

Each sample of the population has a percentage of individuals which has this characteristic. p is the proportion of success of this random variable in the population. The proportion of failure is q = 1 – p

Let all samples of size n in the population. Each sample has a proportion of individuals with this characteristic.

The distribution associated to the random variable that matches each sample with its proportion is called sampling distribution of a proportion.

As for big populations the binomial distribution approximates a normal one, the sampling distribution of a proportion follows a normal distribution, too if n is greater enough, n ≥ 30, and np ≥ 5, nq ≥ 5

Because generally the proportions of the population are unknown, then we approximate it by the sample ones.

Example: A machine makes precision parts. Generally 3% of the parts it makes are defective. A customer receives a 500 part box.

a) What is the probability that he will find more than 5% of the parts in the box are defective?

b) What is the probability that he will find fewer than 1% of defective parts?  Exercise: Suppose the proportion of all college students who have smoked in the past 6 months is p=0.40. For a class of n=200 that is representative of the population of all students, what is the probability that the proportion of students who have smoked in the past 6 months is less than 0.32?

Solution: 0.0104