Could you please solve the problem below? I need detailed mathematical process steps and explanation. The level of nitrogen oxides $($NOX$)$ in the exhaust of a particular car model varies with mean $0\mathrm{.}9$ grams per mile and standard deviation $0\mathrm{.}2$ grams per mile $.$

$($a$)$ What sample size is needed so that the standard deviation of the sampling distribution is $0\mathrm{.}01$ grams per mile $?$

ANSWER: $?$

$($b$)$ If a smaller sample is considered, the standard deviation for $\backslash$bar x would be $?$

$.($NOTE: Enter $\text{'}\text{'}$SMALLER$\text{'}\text{'},\text{'}\text{'}$LARGER$\text{'}\text{'}$ or $\text{'}\text{'}$THE SAME'$\text{'}$ without the quotes.$)$