Q. 7. 47 - Elementary Statistics

Question

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$400$.

Step-by-Step Solution

Answer

(a) The population is made up of all newborn newborns, and the population variable is their weight.

(b) The mean is $3369\mathrm{gm}$ and the standard deviation is$41.08\mathrm{gm}$.

(c) The Mean is $3369\mathrm{gm}$ and the standard deviation is $29.05\mathrm{gm}$.

Part (a) Step 1: Given information

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

Part (a) Step 2: Explanation

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.

Part (b) Step 1: Given information

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.

Part (b) Step 2: Explanation

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:$\sigma =581 gm$
Then the sample size is$n=200$
So,

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

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

$=41.08\mathrm{gm}$

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

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

Part (c) Step 1: Given information

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

Part (c) Step 2: Explanation

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 $\mu =3369\mathrm{gm}$

So,

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

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

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