Chapter 14.4, Problem 11E

(a)

To determine

To find:

The interval about the mean that includes $45\%$ of the data.

Answer to Problem 11E

  $79.6\text{to}84.4$

Explanation of Solution

Given:

Normally distributed data is $=82$

Standard deviation $=4$

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 $=82$

Standard deviation $\left(79.6\text{to}84.4\right)$

  $\sigma =4$

Using the formula

  $\begin{array}{l}P=.275\\ z_1=-.6\\ -.6=\frac{x-82}{4}\\ x=79.6\\ =82-79.6\\ =2.4\\ =82+2.4\\ =84.4\end{array}$

The interval about the mean is $79.6\text{to}84.4$

(b)

To determine

To find:

The interval about the mean that includes $80\%$ of the data.

Answer to Problem 11E

  $76.86\text{to}87.14$

Explanation of Solution

Given:

Normally distributed data is $=82$

Standard deviation $=4$

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 $=82$

Standard deviation $=4$

  $\sigma =4$

Using the formula

  $\begin{array}{l}P=.1\\ z_1=-1.285\\ -1.285=\frac{x-82}{4}\\ x=76.86\\ =82-76.86\\ =5.14\\ =82+5.14\\ =87.14\end{array}$

The interval about the mean is $76.86\text{to}87.14$

(c)

To determine

To find:

The percent of the data is between $76$ and $88$ .

Answer to Problem 11E

  $86.638\%$

Explanation of Solution

Given:

Normally distributed data is $=82$

Standard deviation $=4$

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 $=82$

Standard deviation $=4$

  $\sigma =4$

Using the formula

  $\begin{array}{l}{z}_1=\frac{88-82}{4}\\ z_1=1.5\\ P_1=0.93319\\ z_2=\frac{76-82}{4}\\ z_2=-1.5\\ P_2=0.06681\\ P=P_1-P_2\\ P=0.93319-0.06681\\ P=86.638\%\end{array}$

The percent of the data is between $76$ and $88$ is $86.638\%$ .

(d)

To determine

To find:

The percent of the data is between $80.5$ and $83.5$ .

Answer to Problem 11E

  $29.24370\%$

Explanation of Solution

Given:

Normally distributed data is $=82$

Standard deviation $=4$

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 $=82$

Standard deviation $=4$

  $\sigma =4$

Using the formula

  $\begin{array}{l}{z}_1=\frac{83.5-82}{4}\\ z_1=.375\\ P_1=0.64617\\ z_2=\frac{80.5-82}{4}\\ z_2=-.375\\ P_2=0.35383\\ P=P_1-P_2\\ P=0.64617-0.35383\\ P=29.24370\%\end{array}$

The percent of the data is between $80.5$ and $83.5$ is $29.24370\%$ .