Question

A mid-size city must decide whether or not to build a new combined bus and train station. To build the new station will require an increase in city taxes. According to a city politician, 70% of all city residents support the tax increase to build a combined bus and train station. An opinion poll of 400 city residents will ask whether they favor a rise in taxes to pay for a combined bus and train station. What is the standard deviation of the distribution of sample proportions?

Answer 1

The standard deviation of the distribution of sample proportions is 0.0229.

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)}`

70% of all city residents support the tax increase to build a combined bus and train station.

This means that
`p = 0.7`

400 city residents

This means that
`n = 400`

What is the standard deviation of the distribution of sample proportions?

By the Central Limit Theorem:

`s = \sqrt{(p(1-p))/(n)} = \sqrt{(0.7*0.3)/(400)} = 0.0229`

The standard deviation of the distribution of sample proportions is 0.0229.