Chapter 14.4, Problem 6CFU

(a)

To determine

To draw:

The graph of a curve that represents the frequency distribution.

Explanation of Solution

Given:

Normally distributed data is $=550$

Standard deviation $=35$

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 $=550$

Standard deviation $=35$

  $\sigma =35$

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 percent of the data is between $515$ and $585$ .

Answer to Problem 6CFU

  $68.3\%$

Explanation of Solution

Given:

Normally distributed data is $=550$

Standard deviation $=35$

Concept used:

Standard deviation of a set is falls within the interval of

  $\left(m-\sigma ,m+\sigma \right)$

Calculation:

Normally distributed data is $=550$

Standard deviation $=35$

  $\sigma =35$

  $68.3\%$ 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)

  $\begin{array}{l}=\left(m-\sigma ,m+\sigma \right)\\ =\left(550-35,550+35\right)\\ =\left(515,583\right)\end{array}$

The percent of the data is between $515$ and $585$ is $68.3\%$ .

(c)

To determine

To find:

The interval about the mean that includes $99.7\%$ of the data.

Answer to Problem 6CFU

  $\left(445,655\right)$

Explanation of Solution

Given:

Normally distributed data is $=550$

Standard deviation $=35$

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 $=550$

Standard deviation $=35$

  $\sigma =35$

  $99.7\%$ of the data lie within $3$ Standard deviation of a set is falls within the interval of

  $\left(m-3\sigma ,m+3\sigma \right)\mathrm{..................}\left(1\right)$

Using standard deviation in equation (1)

  $\begin{array}{l}=\left(m-3\sigma ,m+3\sigma \right)\\ =\left(550-3\times 35,550+3\times 35\right)\\ =\left(445,655\right)\end{array}$

The interval about the mean that includes $99.7\%$ of the data is $\left(445,655\right)$

(d)

To determine

To find:

The value of $200$ are in set of data isbetween $480$ and $620$ .

Answer to Problem 6CFU

  $191$

Explanation of Solution

Given:

Normally distributed data is $=550$

Standard deviation $=35$

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 $=550$

Standard deviation $=35$

  $\sigma =35$

Each of the limits in terms of the mean

  $\begin{array}{l}550-480=70\\ 620-550=70\end{array}$

Using the formula

  $\begin{array}{l}\overline{X}\pm t\sigma =550\pm 70\\ t\sigma =70\\ t\left(35\right)=70\\ t=\frac{70}{35}\\ t=2\end{array}$

  $t=2\to 95.5\%$

The values lie between $480$ and $620$

The number of values

  $\begin{array}{l}=95.5\%\\ =0.955\end{array}$

  $\begin{array}{l}=0.995\times 121\\ =191\end{array}$

The value of $200$ are in set of data is between $480$ and $620$ is $191$ .