Answer :
To determine the standard deviation of the distribution of sample means, we utilize the formula for the standard deviation of the sample mean, which is derived from the properties of a normally distributed population.
The formula to find the standard deviation of the distribution of sample means (also known as the standard error) is:
[tex]\sigma_{\bar{x}} = \frac{\sigma}{\sqrt{n}}[/tex]
where:
- [tex]\sigma_{\bar{x}}[/tex] is the standard deviation of the sample means.
- [tex]\sigma[/tex] is the population standard deviation.
- [tex]n[/tex] is the sample size.
In this particular problem:
- The population standard deviation ([tex]\sigma[/tex]) is given as 30.3.
- The sample size ([tex]n[/tex]) is given as 132.
Substitute these values into the formula:
[tex]\sigma_{\bar{x}} = \frac{30.3}{\sqrt{132}}[/tex]
Calculate [tex]\sqrt{132}[/tex]:
[tex]\sqrt{132} \approx 11.49[/tex]
Now, divide the population standard deviation by the square root of the sample size:
[tex]\sigma_{\bar{x}} = \frac{30.3}{11.49} \approx 2.64[/tex]
Therefore, the standard deviation of the distribution of sample means is approximately 2.64. This tells you how much the sample means are expected to vary from the actual population mean when you draw a sample of size 132 from this population.
