Chapter 6.3, Problem 40P

(a)

To determine

Find the probability that the cycle time would exceed 60 minutes, given that it has exceeded 50 minutes.

Answer to Problem 40P

The probability that the cycle time would exceed 60 minutes, given that it has exceeded 50 minutes is 0.3132.

Explanation of Solution

Calculation:

Z score:

The number of standard deviations the original measurement x is from the value of mean $\mu$ is measured using the z-score or z value. The formula for z score is,

$z=\frac{x-\mu }{\sigma }$

In the formula, x is the raw score, $\mu$ is the mean and $\sigma$ is the standard deviation.

The conditional probability formula is,

$P\left(A|B\right)=\frac{P\left(A\text{ and }B\right)}{P\left(B\right)}$

The variable x denotes the cycle time in minutes for the truck to leave the construction site.

The probability that the cycle time would exceed 60 minutes, given that it has exceeded 50 minutes is,

$\begin{array}{c}P\left(x>60|x>50\right)=\frac{P\left(x>60\text{ and }x>50\right)}{P\left(x>50\right)}\\ =\frac{P\left(x>60\right)}{P\left(x>50\right)}\end{array}$

For $P\left(x>60\right)$:

Substitute x as 60, $\mu$ as 45 and $\sigma$ as 12 in the formula of z score.

$\begin{array}{c}z=\frac{60-45}{12}\\ =\frac{15}{12}\\ =1.25\end{array}$

Use the Appendix II: Tables, Table 5: Areas of a Standard Normal Distribution: to obtain probability less than 1.25.

• Locate the value 1.2 in column z.
• Locate the value 0.05 in top row.
• The intersecting value of row and column is 0.8944.

The probability is,

$\begin{array}{c}P\left(x>60\right)=P\left(z>1.25\right)\\ =1-P\left(z<1.25\right)\\ =1-0.8944\\ =0.1056\end{array}$

For $P\left(x>50\right)$:

Substitute x as 50, $\mu$ as 45 and $\sigma$ as 12 in the formula of z score.

$\begin{array}{c}z=\frac{50-45}{12}\\ =\frac{5}{12}\\ =0.42\end{array}$

Use the Appendix II: Tables, Table 5: Areas of a Standard Normal Distribution: to obtain probability less than 0.42.

• Locate the value 0.4 in column z.
• Locate the value 0.02 in top row.
• The intersecting value of row and column is 0.6628.

The probability is,

$\begin{array}{c}P\left(x>50\right)=P\left(z>0.42\right)\\ =1-P\left(z<0.42\right)\\ =1-0.6628\\ =0.3372\end{array}$

Substituting 0.1056 for $P\left(x>60\right)$ and 0.3372 for $P\left(x>50\right)$ in the required probability:

$\begin{array}{c}P\left(x>60|x>50\right)=\frac{0.1056}{0.3372}\\ =0.3132\end{array}$

Hence, the probability that the cycle time would exceed 60 minutes, given that it has exceeded 50 minutes is 0.3132.

(b)

To determine

Find the probability that the cycle time would exceed 55 minutes, given that it has exceeded 40 minutes.

Answer to Problem 40P

The probability that the cycle time would exceed 55 minutes, given that it has exceeded 40 minutes is 0.3067.

Explanation of Solution

Calculation:

The probability that the cycle time would exceed 55 minutes, given that it has exceeded 40 minutes is,

$\begin{array}{c}P\left(x>55|x>40\right)=\frac{P\left(x>55\text{ and }x>40\right)}{P\left(x>40\right)}\\ =\frac{P\left(x>55\right)}{P\left(x>40\right)}\end{array}$

For $P\left(x>55\right)$:

Substitute x as 55, $\mu$ as 45 and $\sigma$ as 12 in the formula of z score.

$\begin{array}{c}z=\frac{55-45}{12}\\ =\frac{10}{12}\\ =0.83\end{array}$

Use the Appendix II: Tables, Table 5: Areas of a Standard Normal Distribution: to obtain probability less than 0.83.

• Locate the value 0.8 in column z.
• Locate the value 0.03 in top row.
• The intersecting value of row and column is 0.7967.

The probability is,

$\begin{array}{c}P\left(x>55\right)=P\left(z>0.83\right)\\ =1-P\left(z<0.83\right)\\ =1-0.7967\\ =0.2033\end{array}$

For $P\left(x>40\right)$:

Substitute x as 40, $\mu$ as 45 and $\sigma$ as 12 in the formula of z score.

$\begin{array}{c}z=\frac{40-45}{12}\\ =\frac{-5}{12}\\ =-0.42\end{array}$

Use the Appendix II: Tables, Table 5: Areas of a Standard Normal Distribution: to obtain probability less than –0.42.

• Locate the value –0.4 in column z.
• Locate the value 0.02 in top row.
• The intersecting value of row and column is 0.3372.

The probability is,

$\begin{array}{c}P\left(x>40\right)=P\left(z>-0.42\right)\\ =1-P\left(z<-0.42\right)\\ =1-0.3372\\ =0.6628\end{array}$

Substituting 0.2033 for $P\left(x>55\right)$ and 0.6628 for $P\left(x>40\right)$ in the required probability:

$\begin{array}{c}P\left(x>55|x>40\right)=\frac{0.2033}{0.6628}\\ =0.3067\end{array}$

Hence, the probability that the cycle time would exceed 55 minutes, given that it has exceeded 40 minutes is 0.3067.