Making auto parts A grinding machine in an auto parts plant prepares axles with a target diameter $\mu =40.125$millimeters (mm). The machine has some variability, so the standard deviation of the diameters is $\sigma =0.002\mathrm{mm}$. The machine operator inspects a random sample of 4 axles each hour for quality control purposes and records the sample mean diameter ${\mathrm{x}}^{-}\overline{x}$. Assume the machine is working properly.

a. Identify the mean of the sampling distribution of ${\mathrm{x}}^{-}·\overline{x}$.

b. Calculate and interpret the standard deviation of the sampling distribution of ${\mathrm{x}}^{-}\overline{x}$.

The values are

The sample mean's sampling distribution mean is equal.

${\mu }_x=\mu =40.125$

The sample as the units of the population mean are the units of the mean of the sampling distribution of the sample mean, and so the mean is $40.125\mathrm{mm}$

The values are,

$$\begin{gathered} \mu =40.125 \\ \sigma =0.002 \\ n=4 \end{gathered}$$

Let use the following formula

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

The sample mean is the mean of the sampling distribution.

${\mu }_{\overline{x}}=\mu =40.125$

The standard deviation of the sample mean's sampling distribution is

${\sigma }_x=\frac{\sigma }{\sqrt{n}}=\frac{0.002}{\sqrt{4}}=\frac{0.002}{2}=0.001$

The units of the sampling distribution's standard deviation of the sample mean are the same as the units of the population standard deviation, so the standard deviation is $0.001\mathrm{mm}$.

In an hour, the sample mean diameter of four randomly selected axles fluctuates by an average of $0.001\mathrm{mm}$from the mean diameter of $40.125$