Chapter 6, Problem 63UP

To determine

Check whether the given statement is true or false.

Answer to Problem 63UP

False.

Explanation of Solution

Sampling distribution of $\overline{x}$:

When a random sample of size n is selected from a population that is normally distributed, then the sampling distribution of $\overline{x}$ is also normally distributed.

Properties:

• The mean of sampling distribution is equal to the mean of the population.

  ${\mu }_{\overline{x}}=E\left(x\right)=\mu$.

• The standard deviation of the sampling distribution is equal to the ratio of the population standard deviation and square root of sample size.

  ${\sigma }_{\overline{x}}=\frac{\sigma }{\sqrt{n}}$.

• When the sample size (n) is large relative to the population size (N), then $\frac{\sigma }{\sqrt{n}}$ must be multiplied by the finite correction factor $\sqrt{\frac{\left(N-n\right)}{\left(N-1\right)}}$.
• For most sampling situations, the correction factor is close to 1 and is ignored.
• The standard deviation ${\sigma }_{\overline{x}}$ is sometimes referred as the standard error of the mean.

From the properties, it is observed that the mean of the sampling distribution of $\overline{x}$ $\left(={\mu }_{\overline{x}}\right)$ is equal to the mean of the population from which the sample is selected, but not the sample mean $\overline{x}$.

Therefore, the given statement “The sample mean $\overline{x}$ will always be equal to ${\mu }_{\overline{x}}$” is false.