Could you please solve the problem below? I need detailed mathematical process steps and explanation. $($a$)$ The length of human pregnancies from conception to birth varies according to a distribution that is approximately Normal with mean $266$ days and standard deviation $16$ days. The probability that the average pregnancy length for $6$ randomly chosen women exceeds $270$ days is about

A$.0\mathrm{.}4$

B$.0\mathrm{.}27$

C$.0\mathrm{.}07$

D$.$ None of the above.

$($b$)$ Scores on the mathematics part of the SAT exam in a recent year were roughly Normal with mean $562$ and standard deviation $112.$ You choose an SRS of $100$ students and average their SAT math scores. If you do this many times, with a sample of size $100$ each time, the mean of the average scores will get close to

A$.5\mathrm{.}62$

B$.562$

C$.56\mathrm{.}2$

D$.$ None of the above.

$($c$)$ A newborn baby has extremely low birth weight $($ELBW$)$ if it weighs less than $1000$ grams. A study of the health of such children in later years examined a random sample of $219$ children. Their mean weight at birth was $\backslash$bar $\{$x$\}=810$ grams. This sample mean is an unbiased estimator of the mean weight $\backslash$mu in the population of all ELBW babies. This means that

A$.$ the average sample mean $\backslash$bar $\{$x$\},$ over all possible samples, is equal to $\backslash$mu$.$

B$.$ the sample mean $\backslash$bar $\{$x$\}$ is always equal to $\backslash$mu$.$

C$.$ the sample mean $\backslash$bar $\{$x$\}$ will have a distribution that is close to Normal.

D$.$ None of the above.