Chapter 7.3, Problem 56E

To determine

To find: the number of axles need to sample if the standard deviation of the sampling distribution of $\overline{x}$ to be 0.0005 mm.

Answer to Problem 56E

16

Explanation of Solution

Given:

  $\begin{array}{c}\mu =40.125\text{mm}\\ \sigma \text{=0}\text{.002mm}\\ {\sigma }_{\overline{x}}=0.0005\text{mm}\end{array}$

Formula used:

Calculation:

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

  ${\text{σ}}_{\text{<mover accent="true"><mo> x</mo> <mo>¯</mo></mover>}}=\frac{\sigma }{\sqrt{n}}$

Multiplying both the side by $\sqrt{n}$

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

Dividing both the side by ${\text{σ}}_{\text{<mover accent="true"><mo> x</mo> <mo>¯</mo></mover>}}$

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

Squaring both the sides

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

Calculating the value of n

  $n=\frac{{\sigma }^2}{\sigma \frac{2}{\overline{x}}}=\frac{{0.002}^2}{{0.0005}^2}=16$

So the sample should be 16 axles if the standard deviation of the sampling distribution of the sample means to be 0.0005 mm.