Chapter 14.4, Problem 14E

(a)

To determine

To find:

The percent of the data is between $6.5$ and $7$ .

Answer to Problem 14E

  $\left(10.5,13.5\right)$

Explanation of Solution

Given:

Normally distributed data is $=6$

Standard deviation $=0.35$

Concept used:

Standard deviation of a set is falls within the interval of

  $\left(m-\sigma ,m+\sigma \right)$

Calculation:

Normally distributed data is $=6$

Standard deviation $=0.35$

  $\sigma =0.35$

Each of the limits in terms of the mean

  $\begin{array}{l}6.5-6=.5\\ 7-6.5=.5\end{array}$

Using the formula

  $\begin{array}{l}\overline{X}\pm t\sigma =6\pm .5\\ t\sigma =.5\\ t\left(0.35\right)=.5\\ t=\frac{1.05}{0.35}\\ t=3\end{array}$

  $t=3\to 88.7\%$

The values lie between $6.5$ and $7$ is $88.7\%$ .

(c)

To determine

To find:

The data is between $5.5$ and $6.2$ .

Answer to Problem 14E

  $88.7\%$

Explanation of Solution

Given:

Normally distributed data is $=6$

Standard deviation $=0.35$

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 $=6$

Standard deviation $=0.35$

  $\sigma =0.35$

Each of the limits in terms of the mean

  $\begin{array}{l}6-5.5=.5\\ 6.2-6=.2\end{array}$

Using the formula

  $\begin{array}{l}\overline{X}\pm t\sigma =6\pm .5\\ t\sigma =.2\\ t\left(0.35\right)=.2\\ t=\frac{.2}{0.35}\\ t=3\end{array}$

  $t=3\to 88.7\%$

The values lie between $6.5$ and $7$ is $88.7\%$ .

(d)

To determine

To find:

The interval about the mean that includes $90\%$ of the data.

Answer to Problem 14E

  $\left(5.65,6.35\right)$

Explanation of Solution

Given:

Normally distributed data is $=6$

Standard deviation $=0.35$

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 $=6$

Standard deviation $=0.35$

  $\sigma =0.35$

  $90\%$ 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)

  $\begin{array}{l}=\left(m-\sigma ,m+\sigma \right)\\ =\left(6-0.35,6+0.35\right)\\ =\left(5.65,6.35\right)\end{array}$

The interval about the mean that includes $90\%$ of the data is $\left(5.65,6.35\right)$