Question

Explain the difference between \(\mu\) and \(\mu_{\bar{x}}\)

Short answer

The key difference between \(\mu\) and \(\mu_{\bar{x}}\) is that \(\mu\) is the population mean and \(\mu_{\bar{x}}\) is the mean of sample means. The former represents the average of the entire population, while the latter represents the average of means of various samples.

Step-by-step solution

Define \(\mu\)

\(\mu\) is the symbol used to represent the population mean in statistics. The population mean is the average of a group as a whole. For example, if you want to know the average height of all people in a city, you would calculate the population mean.

Define \(\mu_{\bar{x}}\)

\(\mu_{\bar{x}}\) is the symbol used to represent the mean of sample means. It is calculated by taking several samples from a population and finding their means, and then taking the average of those means. This can also be called the expected value of the sample mean.

Explain the difference

The main difference between \(\mu\) and \(\mu_{\bar{x}}\) is that \(\mu\) represents the average of all individuals in a population while \(\mu_{\bar{x}}\) represents the average of several sample means taken from the population. \(\mu\) is used when the entire population can be measured. \(\mu_{\bar{x}}\) is used when only samples of the population can be obtained.