Chapter 11.1, Problem 28E

a.

To determine

To find the percent of the plants that are taller than $16$ inches.

Explanation of Solution

Given:

The heights of the plants are normally distributed with a mean of $12$ inches and a standard deviation of $2$ inches.

Calculation:

Using the formula

  $\begin{array}{l} \mathrm{X}=\frac{\mathrm{x}-\mathrm{m}}{\text{ standard deviation}}\\ \mathrm{X}=\frac{16-12}{2}\\ \mathrm{X}=2\end{array}$

Finding $2$ on standard deviation chart,

  $1-0.9772=0.228$ [Finding the percent on the right]

So, the percent of the plants that are taller than $16$ inches are $22.8\mathrm{\%}$

b.

To determine

To find the percent of the plants that are at most $13$ inches.

Explanation of Solution

Given:

The heights of the plants are normally distributed with a mean of $12$ inches and a standard deviation of $2$ inches.

Calculation:

Using the formula

  $\begin{array}{l} \mathrm{X}=\frac{\mathrm{x}-\mathrm{m}}{\text{ standard deviation}}\\ \mathrm{X}=\frac{16-12}{2}\\ \mathrm{X}=2\end{array}$

Finding $2$ on standard deviation chart,

  $1-0.9772=0.228$ [Finding the percent on the right]

So, the percent of the plants that are taller than $16$ inches are $22.8\mathrm{\%}$

c.

To determine

To find the percent of the plants that are between $7$ inches and $14$ inches.

Explanation of Solution

Given:

The heights of the plants are normally distributed with a mean of $12$ inches and a standard deviation of $2$ inches.

Calculation:

Using the formula

  $\begin{array}{l} \mathrm{X}=\frac{\mathrm{x}-\mathrm{m}}{\text{ standard deviation}}\\ \mathrm{X}=\frac{16-12}{2}\\ \mathrm{X}=2\end{array}$

Finding $2$ on standard deviation chart,

  $1-0.9772=0.228$ [Finding the percent on the right]

So, the percent of the plants that are taller than $16$ inches are $22.8\mathrm{\%}$

d.

To determine

To find the percent of the plants that are at least $3$ inches taller than or at least $3$ inches shorter than the mean height.

Explanation of Solution

Given:

The heights of the plants are normally distributed with a mean of $12$ inches and a standard deviation of $2$ inches.

Calculation:

Using the formula

  $\begin{array}{l} \mathrm{X}=\frac{\mathrm{x}-\mathrm{m}}{\text{ standard deviation}}\\ \mathrm{X}=\frac{16-12}{2}\\ \mathrm{X}=2\end{array}$

Finding $2$ on standard deviation chart,

  $1-0.9772=0.228$ [Finding the percent on the right]

So, the percent of the plants that are taller than $16$ inches are $22.8\mathrm{\%}$