91Ó°ÊÓ

Given in the question that,

Mean $=40.125mm$

Standard deviation$=0.002mm$

Number of axles$=4$we have to find that how many axles would you need to sample if you wanted the standard deviation of the sampling distribution $\stackrel{¯}{x}$to be$0.0005$.

The mean of the sampling distribution of $\stackrel{Áåž}{p}is{\mathrm{μ}}_{\stackrel{Áåž}{p}}=p$

The standard deviation of a sample distribution of the sample mean $\stackrel{¯}{x}$if the population standard deviation is can be written as,

${\mathrm{σ}}_{\stackrel{¯}{x}}=\frac{\mathrm{σ}}{\sqrt{n}}$

Here, number of axles is the sample size.

Also,

Population standard deviation,$\mathrm{σ}=0.002$

Standard deviation of the sample,${\mathrm{σ}}_{\stackrel{¯}{x}}=0.0005$

It is known that the standard deviation of a sample distribution of the sample mean if the population standard deviation is can be written as,

${\mathrm{σ}}_{\stackrel{¯}{x}}=\frac{\mathrm{σ}}{\sqrt{n}}$

To find the sample size, use the above formula.

Write the formula as:

$\sqrt{n}=\frac{\mathrm{σ}}{{\mathrm{σ}}_{\stackrel{¯}{x}}}$

squares both side

$n=\frac{{\mathrm{σ}}^2}{{\mathrm{σ}}_{\stackrel{¯}{x}}2}$

substitute $0.002for\mathrm{σ}and0.0005for{\mathrm{σ}}_{\stackrel{¯}{x}}$

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

hence the required number of axles to sample is$16$