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?

The Central Limit Theorem estabilishes that, for a normally distributed random variable X, with mean [tex]\mu[/tex] and standard deviation [tex]\sigma[/tex], the sampling distribution of the sample means with size n can be approximated to a normal distribution with mean [tex]\mu[/tex] and standard deviation [tex]s = \frac{\sigma}{\sqrt{n}}[/tex].

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 [tex]\mu = p[/tex] and standard deviation [tex]s = \sqrt{\frac{p(1-p)}{n}}[/tex]

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

This means that [tex]p = 0.7[/tex]

400 city residents

This means that [tex]n = 400[/tex]

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

By the Central Limit Theorem:

[tex]s = \sqrt{\frac{p(1-p)}{n}} = \sqrt{\frac{0.7*0.3}{400}} = 0.0229[/tex]

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