What is the variance of a random sample of size 5 that is taken with replacement from a population with mean = 15 and  variance = 10?

1. Break Down the Problem

To find the variance of a random sample size of 5 drawn with replacement from a population, we need to understand how the sample variance relates to the population variance.

2. Relevant Concepts

The variance of a random sample drawn from a population with known variance can be calculated using the formula for the variance of the sample mean: $\sigma^2_{\bar{x}} = \frac{\sigma^2}{n}$ where:

• $\sigma^2_{\bar{x}}$ is the variance of the sample mean,
• $\sigma^2$ is the population variance,
• $n$ is the sample size.

3. Analysis and Detail

Given:

• Population variance ($\sigma^2$) = 10
• Sample size ($n$) = 5

We can plug these values into the formula: $\sigma^2_{\bar{x}} = \frac{10}{5}$

Calculating this gives: $\sigma^2_{\bar{x}} = 2$

4. Verify and Summarize

We calculated the variance of the sample mean, which is 2. This indicates how much the sample mean $\bar{x}$ varies from the population mean.

Final Answer

The variance of the random sample of size 5 is $2$.