Q. 50 - The Practice of Statistics for AP

Question

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 $\sigma =0.002$ mm. The machine operator inspects a random sample of $4$ axles each hour for quality control purposes and records the sample mean diameter $\overline{x}$. Assuming that the process is working properly, what are the mean and standard deviation of the sampling distribution of $\overline{x}$? Explain 

Step-by-Step Solution

Answer

The mean of the sampling distribution is ${\mu }_{\overline{x}}=40.125 mm$

The standard deviation of the sampling distribution is ${\sigma }_{\overline{x}}=0.001 mm$

Step 1: Given information

$\sigma =0.002mm$

$\mu =40.125mm$

Samples taken $=4$

Find the mean and standard deviation of the sample.

Step 2: Explanation

The given data is 

$\mu =40.125 mm$

$\sigma =0.002 mm$

$n=4$

The population mean is equal to the mean of the sampling distribution of the sample mean $\overline{x}$

$$\begin{gathered} {\mu }_{\overline{x}}=\mu \\ =40.125mm \end{gathered}$$

The standard deviation of the sampling distribution of the sample mean $\overline{x}$ is calculated by dividing the population standard deviation by the square root of the sample size.

$$\begin{gathered} {\sigma }_{\overline{x}}=\frac{\sigma }{\sqrt{n}} \\ =\frac{0.002}{\sqrt{4}} \\ =0.001mm \end{gathered}$$.