Chapter 1.4, Problem 112E

(a)

To determine

To graph: The normal curve.

Explanation of Solution

Graph: The normal curve can be obtained by using the Minitab software. Follow the steps given below:

Step 1: Enter the data in the Minitab worksheet.

Step 2: Go to Graph and then select the option ‘probability distribution plot.’ Then, select View single. Click on Ok.

Step 3: Enter the value of 150 in mean and 35 in standard deviation, and click on Ok.

The normal curve can be obtained as:

(b)

To determine

To find: The table for values of scores corresponds to the standard deviation from the mean.

Answer to Problem 112E

Solution: The required table can be shown below:

	Lower limit	Upper limit
$150\pm 35$	$115$	$185$
$150\pm \left(2\times 35\right)$	$80$	$220$
$150\pm \left(3\times 35\right)$	$45$	$255$

Explanation of Solution

Calculation: The table corresponding to values of scores for the standard deviation deviating from mean can be seen as done below:

	Lower limit	Upper limit
$\mu \pm \sigma =150\pm 35$	$\begin{array}{c}\mu -\sigma =150-35\\ =115\end{array}$	$\begin{array}{c}\mu +\sigma =150+35\\ =185\end{array}$
$\mu \pm 2\sigma =150\pm \left(2\times 35\right)$	$\begin{array}{c}\mu -2\sigma =150-\left(2\times 35\right)\\ =80\end{array}$	$\begin{array}{c}\mu +2\sigma =150+\left(2\times 35\right)\\ =220\end{array}$
$\mu \pm 3\sigma =150\pm \left(3\times 35\right)$	$\begin{array}{c}\mu -3\sigma =150-\left(3\times 35\right)\\ =45\end{array}$	$\begin{array}{c}\mu +3\sigma =150+\left(3\times 35\right)\\ =255\end{array}$

To determine

To graph: The sketch for the points calculated above.

Explanation of Solution

Graph: The lower limits and upper limits were calculated above. The sketch of these points can be made using the $68-95-99.7$ rule.

The diagram can be seen as below:

(c)

To determine

To find: The range of reading scores using the $68-95-99.7$ rule.

Answer to Problem 112E

Solution: The range of reading scores are $\underset{\_}{\left(115,185\right)}$, $\underset{\_}{\left(80,220\right)}$, and $\underset{\_}{\left(45,225\right)}$.

Explanation of Solution

Calculation: The $68-95-99.7$ rule states that in the normal distribution, 68% of data falls under $\left(\mu \pm \sigma \right)$, 95% of the data falls under $\left(\mu \pm 2\sigma \right)$, and 99.7% of the data falls under $\left(\mu \pm 3\sigma \right)$.

With the given values of mean and standard deviation, the range of reading score value for the students can be obtained by the steps below:

Step 1: For calculating 68% of data,

$\begin{array}{c}\mu \pm \sigma =\left(\mu -\sigma ,\mu +\sigma \right)\\ =\left(150-35,150+35\right)\\ =\left(115,185\right)\end{array}$

Step 2: For calculating 95% of data,

$\begin{array}{c}\mu \pm 2\sigma =\left(\mu -2\sigma ,\mu +2\sigma \right)\\ =\left(150-70,150+70\right)\\ =\left(80,220\right)\end{array}$

Step 3: For calculating 99.7% of data,

$\begin{array}{c}\mu \pm 3\sigma =\left(\mu -3\sigma ,\mu +3\sigma \right)\\ =\left(150-105,150+105\right)\\ =\left(45,225\right)\end{array}$