Q 8.10RP. - Elementary Statistics

Question

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 the birth weights of babies. According to the article, for babies born within the "normal" gestational range of $37-43$ weeks, birth weights are normally distributed with a mean of $3432$ grams ($7$ pounds $9$ ounces) and a standard deviation of $482$ grams ( $1$ pound $1$ ounce). For samples of $15$ such birth weights, identify the distribution of each variable.

a. $\frac{\overrightarrow{x}-3432}{482/\sqrt{15}}$

b. $\frac{\hat{x}-3432}{s/\sqrt{15}}$

Step-by-Step Solution

Answer

Part (a) The standardized version of $\overline{x}$ follows standard normal distribution as birth weight, $x$ follows normal distribution with mean $\mu$ and S.D $\sigma$

Part (b) $14$

Part (a) Step 1: Given information

Birth weights $\left(x\right)$ follows normal distribution with mean $\mu =3432\mathrm{gm}$ and S.D. $\sigma =482$

Sample size, $n=15$

Sample s.d. = $s$

Part (a) Step 2: Concept

The formula used:  Standardized version of $\frac{\overline{x}-\mu }{\frac{\sigma }{\sqrt{n}}},$

Part (a) Step 3: Calculation

$\frac{\overline{x}-\mu }{\frac{\sigma }{\sqrt{n}}}=\frac{\overline{x}-3432}{\frac{482}{\sqrt{15}}}$,

As a birth weight, the standardized form of $\overline{x}$ follows a conventional normal distribution, while $x$ follows a normal distribution with a mean $\mu$ and standard deviation $\sigma$

Part (b) Step 1: Calculation

$\frac{\overline{x}-\mu }{\frac{s}{\sqrt{n}}}=\frac{\overline{x}-3432}{\frac{s}{\sqrt{15}}}$

The studentized version of $\overline{x}$ follows $\mathrm{t}$ - distribution with degrees of freedom

$(n-1)=(15-1)=14$