Chapter 1.4, Problem 105E

(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 288 in mean and 32 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 105E

Solution: The table can be shown below:

	Lower limit	Upper limit
$288\pm 32$	$256$	$320$
$288\pm 2\times 32$	$224$	$352$
$288\pm 3\times 32$	$192$	$384$

Explanation of Solution

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

	Lower limit	Upper limit
$\mu \pm \sigma =288\pm 32$	$\begin{array}{c}\mu -\sigma =288-32\\ =256\end{array}$	$\begin{array}{c}\mu +\sigma =288+32\\ =320\end{array}$
$\mu \pm 2\sigma =288\pm \left(2\times 32\right)$	$\begin{array}{c}\mu -2\sigma =288-\left(2\times 32\right)\\ =224\end{array}$	$\begin{array}{c}\mu +2\sigma =288+\left(2\times 32\right)\\ =352\end{array}$
$\mu \pm 3\sigma =288\pm \left(3\times 32\right)$	$\begin{array}{c}\mu -3\sigma =288-\left(3\times 32\right)\\ =192\end{array}$	$\begin{array}{c}\mu +3\sigma =288+\left(3\times 32\right)\\ =384\end{array}$

To determine

To graph: The sketch for the points calculated above.

Explanation of Solution

The lower limits and upper limits were calculated above.

Graph: The sketch of these points can be made using the $68-95-99.7$ rule.

(c)

To determine

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

Answer to Problem 105E

Solution: The range of reading scores are $\underset{\_}{\left(256,320\right)}$, $\underset{\_}{\left(224,352\right)}$, and $\underset{\_}{\left(192,384\right)}$.

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,

$\begin{array}{c}\mu \pm \sigma =\left(\mu -\sigma ,\mu +\sigma \right)\\ =\left(288-32,288+32\right)\\ =\left(256,320\right)\end{array}$

Step 2: For calculating 95% of data,

$\begin{array}{c}\mu \pm 2\sigma =\left(\mu -2\sigma ,\mu +2\sigma \right)\\ =\left(288-64,288+64\right)\\ =\left(224,352\right)\end{array}$

Step 3: For calculating 99.7% of data,

$\begin{array}{c}\mu \pm 3\sigma =\left(\mu -3\sigma ,\mu +3\sigma \right)\\ =\left(288-96,288+96\right)\\ =\left(192,384\right)\end{array}$