Mathematics: 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?

12. Monthly electric bills for all households in a

Ответ:

kat9940

23.01.2022

Mathematics

$\mu=90$ and $\sigma=25$

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

$\bar X \sim N(\mu, \frac{\sigma}{\sqrt{n}})$

With:

$\mu_{\bar x}= 90$

$\sigma_{\bar X}= \frac{25}{\sqrt{75}}= 2.887$

Step-by-step 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:

$\bar X \sim N(\mu, \frac{\sigma}{\sqrt{n}})$

With:

$\mu_{\bar x}= 90$

$\sigma_{\bar X}= \frac{25}{\sqrt{75}}= 2.887$