Question

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 ${\mathrm{x}}^{-}\overline{x}$.

Short answer

(a). The units of the mean of the sampling distribution of the sample mean are the sample as the units of the population mean and thus the mean is $40.125\mathrm{mm}$

(b). The sample mean diameter of 4 randomly selected axles in an hour varies on average by $0.001\mathrm{mm}$from the mean diameter of $40.125$

Step-by-step solution

part(a) step 1: Given information

The values are

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

part(a) step 2: Calculation

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 $40.125\mathrm{mm}$

Part(b) step 1: Given information 

The values are,

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

Let use the following formula

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

Part(b) step 2: Calculation

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 $40.125$