Question

Single adults: According to a Pew Research Center analysis of census data, in 2012, 20% of American adults ages 25 and older had never been married. (Source: Wang, W., and Parker, K. (2014). Record Share of Americans Have Never Been Married. Pew Research Center.) If we repeatedly obtain random samples of 50 adults, what is the standard deviation of the sampling distribution of sample proportions? Enter your answer in decimal form rounded to two decimal places.

Answer 1

0.06

Explanation:

The Central Limit Theorem estabilishes that, for a normally distributed random variable X, with mean
`\mu` and standard deviation
`\sigma`, the sampling distribution of the sample means with size n can be approximated to a normal distribution with mean
`\mu` and standard deviation
`s = (\sigma)/(√(n))`.

For a skewed variable, the Central Limit Theorem can also be applied, as long as n is at least 30.

For a proportion p in a sample of size n, the standard deviation is
`s = \sqrt{(p(1-p))/(n)}`.

In this problem, we have that:

`p = 0.2, n = 50`

So

`s = \sqrt{(p(1-p))/(n)}`

`s = \sqrt{(0.2*0.8)/(50)}`

`s = 0.06`