Chapter 14.4, Problem 7CFU

(a)

To determine

To find:

The percent of the data is between $22$ and $26$ .

Answer to Problem 7CFU

The percent of the data is between $22$ and $26$ is $68\%$

Explanation of Solution

Given:

Normally distributed data is $=500$

Standard deviation $=2$

The mean $=24$

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 $=500$

Standard deviation $=2$

  $\sigma =2$

Using the formula

  $\begin{array}{l}t\sigma =48\\ t\left(2\right)=48\\ t=24\\ P=68\%\end{array}$

The percent of the data is between $22$ and $26$ is $68\%$ .

(b)

To determine

To find:

The percent of the data is between

  $20.5$ and $27.5$ .

Answer to Problem 7CFU

  $91.988$

Explanation of Solution

Given:

Normally distributed data is $=500$

Standard deviation $=2$

The mean $=24$

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 $=500$

Standard deviation $=2$

  $\sigma =2$

Using the formula

  $\begin{array}{l}{z}_1=\frac{27.5-24}{2}\\ z_1=1.75\\ P_1=0.95994\\ z_2=\frac{20.5-24}{2}\\ z_2=-1.75\\ P_2=0.9006\\ P=P_1-P_2\\ P=0.95994-0.9006\\ P=91.988\end{array}$

The percent of the data is between $20.5$ and $27.5$ is $91.988$ .

(c)

To determine

To find:

The interval about the mean that includes $50\%$ of the data.

Answer to Problem 7CFU

  $22.65\text{to}25.35$

Explanation of Solution

Given:

Normally distributed data is $=500$

Standard deviation $=2$

The mean $=24$

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 $=500$

Standard deviation $=2$

  $\sigma =2$

Using the formula

  $\begin{array}{l}{P}_1=0.75\\ z_1=0.675\\ P_2=0.25\\ z_2=-0.67\\ 0.675=\frac{x_1-24}{2}\\ x_1=25.35\\ -0.675=\frac{x_2-24}{2}\\ x_2=22.65\end{array}$

The interval about the mean that includes $50\%$ of the data is $22.65\text{to}25.35$

(d)

To determine

To find:

The interval about the mean that includes $95\%$ of the data.

Answer to Problem 7CFU

  $20\text{to}28$

Explanation of Solution

Given:

Normally distributed data is $=500$

Standard deviation $=2$

The mean $=24$

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 $=500$

Standard deviation $=2$

  $\sigma =2$

Using the formula

  $\begin{array}{l}{P}_1=0.75\\ z_1=0.675\\ P_2=0.25\\ z_2=-0.67\\ 0.675=\frac{x_1-24}{2}\\ x_1=28\\ -0.675=\frac{x_2-24}{2}\\ x_2=20\end{array}$

The interval about the mean that includes $95\%$ of the data is $20\text{to}28$