Q.7.32 - Elementary Statistics

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.

Step-by-Step Solution

Answer

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

Step 1: Given Information

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

Step 2: 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.