Section B $($Attempt ONE question in this section. $30$ marks$)$

$6.$ You are in charge of a quality control section in a pharmaceutical company. The specifications of a pharmaceutical tablet produced by the company state that the content of the active ingredient needs to be $10\mathrm{.}0mg.$ A random sampling of $250$ tablets revealed that $14$ failed to meet this requirement.

$($a$)$ Use this information to estimate the relative standard deviation for the measurement.

$($b$)$ The quality control specifications of the company require that the overall relative standard deviation of the analysis should not exceed $5\%,$ what is the size of the gross sample? If the relative analytical standard deviation is $3\%,$ how many such samples need to be taken for the laboratory?

$($c$)$ The pharmaceutical tablet has extremely high production costs. Consequently, management considers your sampling needs to be too high. The average mass of the tablet is $165mg.$ It is cylindrical in form with an average diameter of $10mm$ and an average height of $2\mathrm{.}5mm.$ Show, with appropriate calculations, what would you do to prevent the sampling requirement from being greater than $250$ tablets.

$($d$)$ A colleague competing with you for a promotion tries to score brownie points by deriding your solution as too simplistic. He claims that the company should invest in state of the art analytical equipment, which although expensive, will reduce the relative analytical standard deviation to $1\%,$ thereby improving the confidence limits of the analysis. Rebut his arguments, with justification.