91Ó°ÊÓ

When we talk about the population mean, we refer to the average value of a particular characteristic within an entire population. The population mean, represented by the symbol \( \mu \), is a central concept in statistics. It offers a measure of the central tendency of the data.

To calculate the population mean, you sum all the values in the population and then divide by the number of values.

For instance, in our exercise, the population mean is provided directly as \( \mu = 52 \). This means that, for the entire population, the average value is 52. Hence, the population mean shows us this common value found within the population.

The sample mean is a very important statistical measure when working with data from a sample instead of an entire population.
The sample mean, denoted \( \mu_{x} \), is basically the average value of a characteristic within a sample set. This gives us an estimation of the population mean.

Interestingly, when a sample is randomly selected from the population, the sample mean \( \mu_{x} \) is equal to the population mean \( \mu \). From our exercise:

• Population Mean \( \mu=52 \)
• Sample Mean \( \mu_{x}=52 \)

So, here we see that \( \mu_{x} = 52 \) as well. This is because our sample mean is intended to reflect the characteristics of the population.

The standard error helps us understand how much the sample mean \( \mu_{x} \) is expected to vary from the population mean \( \mu \). It's denoted as \( \sigma_{x} \). By calculating the standard error, we can gauge the accuracy of our sample mean as an estimate of the population mean.

The standard error is calculated using the formula:
\[ \sigma_{x} = \frac{\sigma}{\sqrt{n}} \]

where:

• \( \sigma \) is the population standard deviation (10 in this exercise).
• \( \sqrt{n} \) is the square root of the sample size (here, \( \sqrt{21} \)).

Plugging in our values:
\[ \sigma_{x} = \frac{10}{\sqrt{21}} \]
\[........ = \frac{10}{4.5826} \]\[........ \approx 2.18 \]
Thus, the standard error \( \sigma_{x} \) is approximately 2.18. This gives us an idea about the variability of sample means if we were taking multiple samples from the same population.