Question

Making auto parts Refer to Exercise 54 . How many axles would you need to sample if you wanted the standard deviation of the sampling distribution of $\mathrm{x}-\overline{x}$to be $0.0005\mathrm{mm}$ ? Justify your answer.

Short answer

If the sample's standard deviation of the sampling distribution is $0.005nm$, the sample should be 16 axles.

Step-by-step solution

Given information 

Given,

$$\begin{gathered} \mu =40.125\mathrm{mm} \\ \sigma =0.002\mathrm{mm} \\ {\sigma }_{\overline{x}}=0.0005\mathrm{mm} \end{gathered}$$

Calculation

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

${\sigma }_1=\frac{\sigma }{\sqrt{n}}$

By multiplying both sides by $\sqrt{n}$

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

By dividing both the side by ${\sigma }_2$

$\sqrt{n}=\frac{\sigma }{{\sigma }_{\mathrm{I}}}$

By squaring both the sides

$n=\frac{{\sigma }^2}{{\sigma }_{\overline{x}}^2}$

Now Calculate the value of $n$

$n=\frac{{\sigma }^2}{\sigma \frac{2}{I}}=\frac{0.{002}^2}{0.{0005}^2}=16$

If the sample's standard deviation of the sampling distribution is$0.0005mm$ the sample should be 16 axles.