Question

You have seen that the larger the sample size, the smaller the sampling error tends to be in estimating a population means by a sample mean. This fact is reflected mathematically by the formula for the standard deviation of the sample mean: ${\sigma }_i=\sigma /\sqrt{n}$. For a fixed sample size, explain what this formula implies about the relationship between the population standard deviation and sampling error.

Short answer

The standard deviation of the sample mean and the standard deviation of the population is thus directly proportional.

Step-by-step solution

Given Information

${\sigma }_{\overline{x}}=\sigma /\sqrt{n}$ is the formula for the sample mean's standard deviation.

Explanation

The standard deviation of the sample mean determines the degree of sampling error to be expected when a population mean is approximated using a sample mean. The sample size $\left(n\right)$is assumed to be constant. In other words, the denominator in the ${\sigma }_{\overline{x}}$formula remains unchanged.

The standard deviation of the sample mean and the standard deviation of the population are thus directly proportional.