You work in the quality control $($QC$)$ lab at a water treatment facility where you ensure that potential contaminants are below federal and state thresholds. The facility you are working at is interested in buying a new spectrophotometric instrument to test nitrate levels in water. Your team recently sent out the same standard samples for analysis by two different instrument manufacturers. The following table contains the measured signal for a series of $10$ replicate measurements using the known standard on each instrument.

$\backslash$table$[[$Measurement Number,Instrument $1,$Instrument $2($ppm$)],[1,4\mathrm{.}22,4\mathrm{.}70],[2,5\mathrm{.}43,4\mathrm{.}81],[3,4\mathrm{.}57,4\mathrm{.}65],[4,4\mathrm{.}11,4\mathrm{.}68],[5,5\mathrm{.}89,4\mathrm{.}89],[6,5\mathrm{.}02,4\mathrm{.}50],[7,4\mathrm{.}53,4\mathrm{.}55],[8,3\mathrm{.}74,4\mathrm{.}62],[9,4\mathrm{.}83,4\mathrm{.}87],[10,4\mathrm{.}32,4\mathrm{.}78]]$

a$)$ What is the signal to noise ratio for each instrument of this standard sample?

b$)$ If you were given a choice, which instrument would you utilize? Why?

c$)$ How many measurements are required to improve the $\frac{S}{N}$ ratio to $\frac{S}{N}=50$ for instrument number $1?$