## Related questions with answers

Suppose that we will take a random sample of size $n$ from a population having mean $\mu$ and standard deviation $\sigma$. For each of the following situations, find the mean, variance, and standard deviation of the sampling distribution of the sample mean $\bar{x}$:

$\mu=10, \quad \sigma=2, \quad n=25$

Solutions

Verified*Recall :* The mean $\mu_{\bar x}$ is just equal to the mean $\mu$.

The variance $\sigma_{\bar x}^2$ is given by the formula,

$\sigma_{\bar{x}}^2=\frac{\sigma^2}{n}$

and the standard deviation is just the square root of the variance,

$\sigma_{\bar{x}}=\frac{\sigma}{\sqrt{n}}$

If the population mean is $\mu$, and standard deviation is $\sigma$, then the mean and standard deviation of the population of all possible sample means are:

$\mu_{\overline{x}}=\mu\quad\text{and}\quad\sigma_{\overline{x}}=\frac{\sigma}{\sqrt{n}}.$

Then, the variance is

$\sigma_{\overline{x}}^2=\frac{\sigma^2}{n}.$

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