Chapter 13, Problem 20P

a.

Summary Introduction

To calculate: The probability that the outcome will be between $16,800 and $31,200.

Introduction:

Probability:

The likelihood of the occurrence of an event or of a proposition to be true, when expressed numerically or quantitatively, is termed as probability.

Answer to Problem 20P

The probability that the outcome will be between $16,800 and $31,200 is 0.8664.

Explanation of Solution

The calculation of the expected value (Z) for the outcome being equal to or greater than $16,800 is shown below.

$\begin{array}{c}Z=\frac{\text{Lower Range}-\text{Expected Value (X)}}{\text{Standard Deviation (}\sigma \text{)}}\\ =\frac{\$16,800-\$24,000}{\$4,800}\\ =\frac{\left(-\right)\$7,200}{\$4,800}\\ =\left(-\right)1.5\end{array}$

The calculation of the expected value (Z) for the outcome being equal to or lower than $31,200 is shown below.

$\begin{array}{c}Z=\frac{\text{Upper Range}-\text{Expected Value (X)}}{\text{Standard Deviation (}\sigma \text{)}}\\ =\frac{\$31,200-\$24,000}{\$4,800}\\ =\frac{\$7,200}{\$4,800}\\ =1.5\end{array}$

The Z values are positive as well as negative 1.5. Hence, the probability of the outcome being between $16,800 and $31,200 is 0.8664.

b.

Summary Introduction

To calculate: The probability that the outcome will be between $14,400 and $33,600.

Introduction:

Probability:

The likelihood of the occurrence of an event or of a proposition to be true, when expressed numerically or quantitatively, is termed as probability.

Answer to Problem 20P

The probability that the outcome will be between $14,400 and $33,600 is 0.9544.

Explanation of Solution

The calculation of the expected value (Z) for the outcome being equal to or greater than $14,400 is shown below.

$\begin{array}{c}Z=\frac{\text{Lower Range}-\text{Expected Value (X)}}{\text{Standard Deviation (}\sigma \text{)}}\\ =\frac{\$14,400-\$24,000}{\$4,800}\\ =\frac{\left(-\right)\$9,600}{\$4,800}\\ =\left(-\right)2\end{array}$

The calculation of the expected value (Z) for the outcome being equal to or lower than $33,600 is shown below.

$\begin{array}{c}Z=\frac{\text{Upper Range}-\text{Expected Value (X)}}{\text{Standard Deviation (}\sigma \text{)}}\\ =\frac{\$33,600-\$24,000}{\$4,800}\\ =\frac{\$9,600}{\$4,800}\\ =2\end{array}$

The Z values are positive as well as negative 2. Hence, the probability of the outcome being between $14,400 and $33,600 is 0.9544.

c.

Summary Introduction

To calculate: The probability that the outcome will be at least $14,400.

Introduction:

Probability:

The likelihood of the occurrence of an event or of a proposition to be true, when expressed numerically or quantitatively, is termed as probability.

Answer to Problem 20P

The probability that the outcome will be at least $14,400 is 0.9544.

Explanation of Solution

The calculation of the expected value (Z) for the outcome being at least $14,400 is shown below.

$\begin{array}{c}Z=\frac{\text{Lower Range}-\text{Expected Value (X)}}{\text{Standard Deviation (}\sigma \text{)}}\\ =\frac{\$14,400-\$24,000}{\$4,800}\\ =\frac{\left(-\right)\$9,600}{\$4,800}\\ =\left(-\right)2\end{array}$

The expected value is 0.4772 when Z is (+ or -) 2 and 0.5000 when Z is 0. So, the probability of the outcome being at least $14,400 is 0.9772. The graph of this probability is shown below.

d.

Summary Introduction

To calculate: The probability that the outcome will be less than $31,900.

Introduction:

Probability:

The likelihood of the occurrence of an event or of a proposition to be true, when expressed numerically or quantitatively, is termed as probability.

Answer to Problem 20P

The probability that the outcome will be less than $31,900 is 0.9544.

Explanation of Solution

The calculation of the expected value (Z) for the outcome being at least $14,400 is shown below.

$\begin{array}{c}Z=\frac{\text{Upper Range}-\text{Expected Value (X)}}{\text{Standard Deviation (}\sigma \text{)}}\\ =\frac{\$31,900-\$24,000}{\$4,800}\\ =\frac{\$7,900}{\$4,800}\\ =1.65\end{array}$

The expected value is 0.4505 when Z is (+ or -) 1.65 and 0.5000 when Z is 0. So, the probability of the outcome being at least $14,400 is 0.9505. The graph of this probability is shown below.

e.

Summary Introduction

To calculate: The probability that the outcome will be less than $19,200 or greater than $26,400.

Introduction:

Probability:

The likelihood of the occurrence of an event or of a proposition to be true, when expressed numerically or quantitatively, is termed as probability.

Answer to Problem 20P

The probability that the outcome will be less than $19,200 or greater than $26,400 is 0.4672.

Explanation of Solution

The calculation of the expected value (Z) for the outcome being less than $19,200 is shown below.

$\begin{array}{c}Z=\frac{\text{Lower Range}-\text{Expected Value (X)}}{\text{Standard Deviation (}\sigma \text{)}}\\ =\frac{\$19,200-\$24,000}{\$4,800}\\ =\frac{\left(-\right)\$4,800}{\$4,800}\\ =\left(-\right)1\end{array}$

The expected value is 0.3413 when Z is (+ or -) 1 and 0.5000 when Z is 0. So, the probability of the outcome being less than $19,200 is 0.1587 (0.5000 – 0.3413).

The calculation of the expected value (Z) for the outcome being greater than $26,400 is shown below.

$\begin{array}{c}Z=\frac{\text{Upper Range}-\text{Expected Value (X)}}{\text{Standard Deviation (}\sigma \text{)}}\\ =\frac{\$26,400-\$24,000}{\$4,800}\\ =\frac{\$2,400}{\$4,800}\\ =0.5\end{array}$

The expected value is 0.1915 when Z is (+ or -) 1 and 0.5000 when Z is 0. So, the probability of the outcome being at least $14,400 is 0.3085 (0.5000 – 0.1915).

Hence, the probability that the outcome will be less than $19,200 or greater than $26,400 is 0.4672. The graph of this probability is shown below.