Chapter 14.4, Problem 9E

(a)

To determine

To draw:

The graph of a curve that represents the frequency distribution.

Explanation of Solution

Given:

Normally distributed data is $=12$

Standard deviation $=1.5$

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 $=12$

Standard deviation $=1.5$

  $\sigma =1.5$

Draw the graph of the frequency distribution

  

The frequency and probability distribution are identically shaped for a normally distributed data set.

This graph is bell curve centered at the mean /median /mode and symmetric about this point.

(b)

To determine

To find:

The interval about the mean that includes $68.3\%$ of the data.

Answer to Problem 9E

  $\left(10.5,13.5\right)$

Explanation of Solution

Given:

Normally distributed data is $=12$

Standard deviation $=1.5$

Concept used:

Standard deviation of a set is falls within the interval of

  $\left(m-\sigma ,m+\sigma \right)$

Calculation:

Normally distributed data is $=12$

Standard deviation $=1.5$

  $\sigma =35$

  $68.3\%$ Standard deviation of a set is falls within the interval of

  $\left(m-\sigma ,m+\sigma \right)\mathrm{..................}\left(1\right)$

Using standard deviation in equation (1)

  $\begin{array}{l}=\left(m-\sigma ,m+\sigma \right)\\ =\left(10.5,13.5\right)\end{array}$

The interval about the mean that includes $68.3\%$ of the data is $\left(10.5,13.5\right)$

(c)

To determine

To find:

The data is between $7.5$ and $16.5$ .

Answer to Problem 9E

  $88.7\%$

Explanation of Solution

Given:

Normally distributed data is $=12$

Standard deviation $=1.5$

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 $=12$

Standard deviation $=1.5$

  $\sigma =1.5$

Each of the limits in terms of the mean

  $\begin{array}{l}12-7.5=4.5\\ 16.5-12=4.5\end{array}$

Using the formula

  $\begin{array}{l}\overline{X}\pm t\sigma =12\pm 4.5\\ t\sigma =3.5\\ t\left(1.5\right)=4.5\\ t=\frac{4.5}{1.5}\\ t=3\end{array}$

  $t=3\to 88.7\%$

The values lie between $7.5$ and $16.5$ is $88.7\%$ .

(d)

To determine

To find:

The data is between $9$ and $15$ .

Answer to Problem 9E

  $95.5\%$

Explanation of Solution

Given:

Normally distributed data is $=12$

Standard deviation $=1.5$

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 $=12$

Standard deviation $=1.5$

  $\sigma =1.5$

Each of the limits in terms of the mean

  $\begin{array}{l}12-9=3\\ 15-12=3\end{array}$

Using the formula

  $\begin{array}{l}\overline{X}\pm t\sigma =12\pm 3\\ t\sigma =3\\ t\left(1.5\right)=3\\ t=\frac{3}{1.5}\\ t=2\end{array}$

  $t=2\to 95.5\%$

The values lie between $7.5$ and $16.5$ is $95.5\%$ .