Chapter 14.4, Problem 13E

(a)

To determine

To find:

The percent of the data is between $100$ and $150$ .

Answer to Problem 13E

  $66.87140$

Explanation of Solution

Given:

Normally distributed data is $=140$

Standard deviation $=20$

Concept used:

Normal distribution: - A normal distribution is a frequency distribution that often occurs

There is a large number of values in a set of data.

The graph of this distribution is a symmetric, bell-shaped curve.

Calculation:

Normally distributed data is $=140$

Standard deviation $=20$

  $\sigma =20$

Using the formula

  $\begin{array}{l}{z}_1=\frac{150-140}{20}\\ z_1=.5\\ P_1=0.69146\\ z_2=\frac{100-140}{20}\\ z_2=-2\\ P_2=0.02275\\ P=P_1-P_2\\ P=0.69146-0.02275\\ P=66.87140\end{array}$

The percent of the data is between $100$ and $150$ is $66.87140$ .

(b)

To determine

To find:

The percent of the data is between $150$ and $180$ .

Answer to Problem 13E

  $28.57$

Explanation of Solution

Given:

Normally distributed data is $=140$

Standard deviation $=20$

Concept used:

Normal distribution: - A normal distribution is a frequency distribution that often occurs

There is a large number of values in a set of data.

The graph of this distribution is a symmetric, bell-shaped curve.

Calculation:

Normally distributed data is $=140$

Standard deviation $=20$

  $\sigma =20$

Using the formula

  $\begin{array}{l}{z}_1=\frac{150-140}{20}\\ z_1=.5\\ P_1=0.97725\\ z_2=\frac{180-140}{20}\\ z_2=2\\ P_2=0.69146\\ P=P_1-P_2\\ P=0.97725-0.69146\\ P=.28578\\ P=28.57\end{array}$

The percent of the data is between $180$ and $150$ is $28.57$ .

(c)

To determine

To find:

The value that data defines the $75\text{th}$ percentile

Answer to Problem 13E

  $153.5$

Explanation of Solution

Given:

Normally distributed data is $=140$

Standard deviation $=20$

Concept used:

Normal distribution: - A normal distribution is a frequency distribution that often occurs

There is a large number of values in a set of data.

The graph of this distribution is a symmetric, bell-shaped curve.

Calculation:

Normally distributed data is $=140$

Standard deviation $=20$

  $\sigma =20$

Using the formula

  $\begin{array}{l}z=.675\\ .675=\frac{x-140}{20}\\ x-140=.675\times 20\\ x-140=13.5\\ x=13.5+140\\ x=153.5\end{array}$

The value that data defines the $75\text{th}$ percentile is $153.5$ .