7.47 Baby Weight. The paper "Are Babies Normal?" by T. Clemons and M. Pagano (The American Statistician, Vol. 53, No, 4. pp. 298-302) focused on birth weights of babies. According to the article, the mean birth weight is$3369$ grams ($7$ pounds, $6.5$ ounces) with a standard deviation of 581 grams.
a. Identify the population and variable.
b. For samples of size $200$, find the mean and standard deviation of all possible sample mean weights.
c. Repeat part (b) for samples of size$400$.

The mean birth weight is $3369$ grams ($7$pounds, $6.5$ounces) with a standard deviation of $581$grams.

All of the newly born babies make up the population.
The weight of a newborn baby is the population variable.
As a result, the population is made up of all newborn newborns, and the population variable is their weight.

To find the mean and standard deviation of all possible sample mean weights for samples of size $200$. Where the mean birth weight is $3369$grams, and the standard deviation of $581$grams.

The sample size is limited in comparison to the population size.

Standard deviation of all possible sample means is ${\sigma }_X=\frac{\sigma }{\sqrt{n}}$
Where, the standard deviation for the population:localid="1651037316481" $\sigma =581gm$
Then the sample size is$n=200$
So,

${\sigma }_X=\frac{\sigma }{\sqrt{n}}$

$=\frac{581}{\sqrt{200}}$

localid="1651037567796" $=41.08\mathrm{gm}$

Hence, mean of all possible sample means localid="1651037663812" $\mu =3369\mathrm{gm}$

As a result, mean is $3369\mathrm{gm}$and standard deviation is localid="1651037574449" $41.08\mathrm{gm}$.

To determine part (b) for samples of size $400$. Where the mean birth weight is $3369$grams, and the standard deviation of $581$ grams.

Standard deviation of all possible sample mean is ${\sigma }_X=\frac{\sigma }{\sqrt{n}}$

Where the sample size $n=400$, Mean of all possible means $\mu =3369\mathrm{gm}$

${\sigma }_X=\frac{\sigma }{\sqrt{n}}$

$=\frac{581}{\sqrt{400}}$

$=29.05gm$
As a result, Mean is $3369\mathrm{gm}$and standard deviation is$29.05\mathrm{gm}$.