Question

Simple Random Sample of Size 25:

Chances are, since you are studying AP Statistics, you plan to earn at least a bachelor's degree from college. According to the U.S. Census Bureau, 29% of Americans age 25 years or older have already earned a bachelor's or more advanced degree. You decide to randomly select 58 adults over the age of 25 to ask if they have earned at least a bachelor's degree. Describe the sampling distribution of the proportion of respondents who have earned at least a bachelor's degree.

Answer 1

By the Central Limit Theorem, it is approximately normal with mean 0.29 and standard deviation 0.0596

Explanation:

Central Limit Theorem

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 sampling distribution of the sample proportion will be approximately normal with mean
`\mu = p` and standard deviation
`s = \sqrt{(p(1-p))/(n)}`

According to the U.S. Census Bureau, 29% of Americans age 25 years or older have already earned a bachelor's or more advanced degree.

This means that
`p = 0.29`

You decide to randomly select 58 adults over the age of 25 to ask if they have earned at least a bachelor's degree.

This means that
`n = 58`

Describe the sampling distribution of the proportion of respondents who have earned at least a bachelor's degree.

By the Central Limit Theorem, it is approximately normal with mean
`\mu = p = 0.29` and the standard deviation is
`s = \sqrt{(p(1-p))/(n)} = \sqrt{(0.29*0.71)/(58)} = 0.0596`