Chapter 3.2, Problem 47E

a.

To determine

Find the number of new-borns whose weigh was less than 4,105 grams.

Answer to Problem 47E

The number of new-borns whose weigh was less than 4,105 grams are 521 or 522.

Explanation of Solution

Calculation:

The mean birth weight of 621 new born babies is 3,234 grams and the standard deviation is 871 grams. The birth weight data is approximately bell-shaped.

The distribution is bell shaped, therefore empirical rule can be used here.

Empirical Rule:

If a sample with mean $\overline{x}$ and standard deviation s has an approximately bell-shaped histogram them;

• Approximately 68% of the observations lie between $\overline{x}-s$ and $\overline{x}+s$.
• Approximately 95% of the observations lie between $\overline{x}-2s$ and $\overline{x}+2s$.
• Approximately all of the observations lie between $\overline{x}-3s$ and $\overline{x}+3s$.

Here $\overline{x}=3,234$ as the sample mean and the sample standard deviation is 871.

The 1 standard deviation limit is,

$\begin{array}{c}\overline{x}\pm s=3,234\pm 871\\ =\left(2,363,4,105\right)\end{array}$

The 2 standard deviation limit is,

$\begin{array}{c}\overline{x}\pm 2s=3,234\pm \left(2\times 871\right)\\ =3,234\pm 1,742\\ =\left(1,492,4,976\right)\end{array}$

The 3 standard deviation limit is,

$\begin{array}{c}\overline{x}\pm 3s=3,234\pm \left(3\times 871\right)\\ =3,234\pm 2,613\\ =\left(621,5,847\right)\end{array}$

The graph of the limits is given below:

It is known that 50% of the observations lie in both sides of the mean, $\overline{x}$.

By the graph it is clear that there will be $\frac{68}{2}=34\%$ of the data in the limit $\left(\overline{x},\overline{x}+s\right)$.

Therefore, in left side of $\left(\overline{x}+s\right)=4,105$, there will be $50\%+34\%=84\%$ of the observation.

Total number of babies is 621.

84% of the babies is,

$\begin{array}{c}\text{number of babies}=621\times \frac{84}{100}\\ =521.64\end{array}$

Hence, the number of new-borns whose weigh was less than 4,105 grams are 521 or 522.

b.

To determine

Find the number of new-borns whose weigh was greater than 1,492 grams.

Answer to Problem 47E

The number of new-borns whose weigh was greater than 1,492 grams is 605 or 606.

Explanation of Solution

Calculation:

It is known that 50% of the observations lie in both sides of the mean, $\overline{x}$.

From the graph of part (a) it is clear that 95% observations lie between $\left(\overline{x}-2s,\overline{x}+2s\right)$.

Therefore, $\frac{95\%}{2}=47.5\%$ of the observations lie in the interval $\left(\overline{x}-2s,\overline{x}\right)$.

Hence, in the right side of $\left(\overline{x}-2s\right)=1,492$, there will be $47.5\%+50\%=97.5\%$ of observations.

Total number of babies is 621.

97.5% of the babies is,

$\begin{array}{c}\text{number of babies}=621\times \frac{97.5}{100}\\ =605.475\end{array}$

Hence, the number of new-borns whose weigh was greater than 1,492 grams is 605 or 606.

c.

To determine

Find the number of new-borns whose weigh was between 3,234 and 4,976 grams.

Answer to Problem 47E

• The number of new-born whose weigh was between 3,234 and 4,976 grams are 295.

Explanation of Solution

Calculation:

It is known that 50% of the observations lie in both sides of the mean, $\overline{x}$.

From the graph of part (a) it is clear that 95% observations lies between $\left(\overline{x}-2s,\overline{x}+2s\right)$.

Therefore, $\frac{95\%}{2}=47.5\%$ of the observations lie in the interval $\left(\overline{x},\overline{x}+2s\right)$.

Thus, $\left\{\overline{x}\left(=3,234\right),\overline{x}+2s\left(=4,976\right)\right\}$ this interval consists of 47.5% of the observations.

Total number of babies is 621.

47.5% of the babies is,

$\begin{array}{c}\text{number of babies}=621\times \frac{47.5}{100}\\ =294.975\end{array}$

• Hence the number of new-born whose weigh was between 3,234 and 4,976 grams are 295.