Question

In a large population, 62% of the households have cable tv. A simple random sample of 100 households is to be contacted and the sample proportion computed. What is the mean and standard deviation of the sampling distribution of the sample proportions

Answer 1

The mean of the sampling distribution of the sample proportions is 0.62 and the standard deviation is 0.0485.

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)}`

62% of the households have cable tv.

This means that
`p = 0.62`

Sample of 100.

This means that
`n = 100`

What is the mean and standard deviation of the sampling distribution of the sample proportions?

`\mu = p = 0.62`

`s = \sqrt{(p(1-p))/(n)} = \sqrt{(0.62*0.38)/(100)} = 0.0485`

The mean of the sampling distribution of the sample proportions is 0.62 and the standard deviation is 0.0485.