Question

12. Monthly electric bills for all households in a city have a skewed probability distribution with mean $90 and standard deviation $25. A simple random sample of 75 households is taken. (a) What is the approximate shape of the distribution of sample means?

Answer 1

`\mu=90` and
`\sigma=25`

From the central limit theorem (n>30) we know that the distribution for the sample mean
`\bar X` is given by:

`\bar X \sim N(\mu, (\sigma)/(√(n)))`

With:

`\mu_(\bar x)= 90`

`\sigma_(\bar X)= (25)/(√(75))= 2.887`

Explanation:

Previous concepts

Normal distribution, is a "probability distribution that is symmetric about the mean, showing that data near the mean are more frequent in occurrence than data far from the mean".

The Z-score is "a numerical measurement used in statistics of a value's relationship to the mean (average) of a group of values, measured in terms of standard deviations from the mean".

Solution to the problem

Let X the random variable that represent the heights of a population, and for this case we know the following properties for X

Where
`\mu=90` and
`\sigma=25`

From the central limit theorem (n>30) we know that the distribution for the sample mean
`\bar X` is given by:

`\bar X \sim N(\mu, (\sigma)/(√(n)))`

With:

`\mu_(\bar x)= 90`

`\sigma_(\bar X)= (25)/(√(75))= 2.887`