Chapter 7, Problem 91CE

John can take either of two routes (A or B) to LAX airport. At midday on a typical Wednesday the travel time on either route is normally distributed with parameters μ_A = 54 minutes, σ_A = 6 minutes, μ_B = 60 minutes, and σ_B = 3 minutes. (a) Which route should he choose if he must be at the airport in 54 minutes to pick up his spouse? (b) Sixty minutes? (c) Sixty-six minutes? Explain carefully.

To determine

Find the better route for John to reach at the airport in 54 minutes to pick up his spouse.

Answer to Problem 91CE

The better route for John to reach at the airport in 54 minutes to pick up his spouse is route A.

Explanation of Solution

Calculation:

It is given that the time on route A follows normal distribution with mean of 54 minutes, standard deviation of 6 minutes and the time on route B follows normal distribution with mean of 60 minutes, standard deviation of 3 minutes.

Normal distribution:

A continuous random variable X is said to follow normal distribution if the probability density function of X is,

$f\left(x\right)=\frac{1}{\sigma \sqrt{2\pi }}{e}^{-\frac{{\left(x-\mu \right)}^2}{2{\sigma }^2}},-\infty <x<\infty$, with mean $\mu$ and standard deviation $\sigma$.

Assume that the random variable X denotes the time to reach airport.

For route A:

It is given that ${\mu }_A=54\text{minutes}$ and ${\sigma }_A=6\text{minutes}$.

Now, the probability to reach at the airport in 54 minutes using route A, implies that $P\left(X<54\right)$.

Probability value:

Software procedure:

Step-by-step software procedure to obtain probability value using EXCEL is as follows:

• Open an EXCEL file.
• In cell A1, enter the formula “=NORM.DIST(54,54,6,1)”.
• Output using EXCEL software is given below:

Therefore, $P\left(X<54\right)=0.5$.

Thus, the probability to reach at the airport in 54 minutes using route A is 0.5.

For route B:

It is given that ${\mu }_B=60\text{minutes}$ and ${\sigma }_B=3\text{minutes}$.

Now, the probability to reach at the airport in 54 minutes using route B, implies that $P\left(X<54\right)$.

Probability value:

Software procedure:

Step-by-step software procedure to obtain probability value using EXCEL is as follows:

• Open an EXCEL file.
• In cell A1, enter the formula “=NORM.DIST(54,60,3,1)”.
• Output using EXCEL software is given below:

Therefore, $P\left(X<54\right)=0.0228$.

Thus, the probability to reach at the airport in 54 minutes using route B is 0.0228.

Therefore, there is more chance to reach airport by route A than by route B.

Thus, the better route for John to reach at the airport in 54 minutes to pick up his spouse is route A.

To determine

Find the better route for John to reach at the airport in 66 minutes to pick up his spouse.

Answer to Problem 91CE

The better route for John to reach at the airport in 60 minutes to pick up his spouse is route A.

Explanation of Solution

Calculation:

For route A:

It is given that ${\mu }_A=54\text{minutes}$ and ${\sigma }_A=6\text{minutes}$.

Now, the probability to reach at the airport in 66 minutes using route A, implies that $P\left(X<60\right)$.

Probability value:

Software procedure:

Step-by-step software procedure to obtain probability value using EXCEL is as follows:

• Open an EXCEL file.
• In cell A1, enter the formula “=NORM.DIST(60,54,6,1)”.
• Output using EXCEL software is given below:

Therefore, $P\left(X<60\right)=0.8413$.

Thus, the probability to reach at the airport in 60 minutes using route A is 0.8413.

For route B:

It is given that ${\mu }_B=60\text{minutes}$ and ${\sigma }_B=3\text{minutes}$.

Now, the probability to reach at the airport in 60 minutes using route B, implies that $P\left(X<60\right)$.

Probability value:

Software procedure:

Step-by-step software procedure to obtain probability value using EXCEL is as follows:

• Open an EXCEL file.
• In cell A1, enter the formula “=NORM.DIST(60,60,3,1)”.
• Output using EXCEL software is given below:

Therefore, $P\left(X<60\right)=0.5$.

Thus, the probability to reach at the airport in 60 minutes using route B is 0.5.

Therefore, there is more chance to reach airport by route A than by route B.

Thus, the better route for John to reach at the airport in 60 minutes to pick up his spouse is route A.

To determine

Find the better route for John to reach at the airport in 60 minutes to pick up his spouse.

Answer to Problem 91CE

The better route for John to reach at the airport in 66 minutes to pick up his spouse is route B.

Explanation of Solution

Calculation:

Coefficient of variation:

Coefficient of variation for a random variable X is defined as,

$\text{CV}=\frac{s}{\overline{x}}\times 100\%$ , where s is the sample standard deviation and $\overline{x}$ is the same mean of X.

It is better to use a random variable with lower CV.

For route A:

It is given that ${\mu }_A=54\text{minutes}$ and ${\sigma }_A=6\text{minutes}$.

Now, the probability to reach at the airport in 66 minutes using route A, implies that $P\left(X<66\right)$.

Probability value:

Software procedure:

Step-by-step software procedure to obtain probability value using EXCEL is as follows:

• Open an EXCEL file.
• In cell A1, enter the formula “=NORM.DIST(66,54,6,1)”.
• Output using EXCEL software is given below:

Therefore, $P\left(X<66\right)=0.9772$.

Thus, the probability to reach at the airport in 66 minutes using route A is 0.9972.

The CV for route A is,

$\begin{array}{c}{\text{CV}}_A=\frac{6}{54}\times 100\%\\ =0.1111\times 100\%\\ =11.11\%\end{array}$

For route B:

It is given that ${\mu }_B=60\text{minutes}$ and ${\sigma }_B=3\text{minutes}$.

Now, the probability to reach at the airport in 66 minutes using route B, implies that $P\left(X<66\right)$.

Probability value:

Software procedure:

Step-by-step software procedure to obtain probability value using EXCEL is as follows:

• Open an EXCEL file.
• In cell A1, enter the formula “=NORM.DIST(66,60,3,1)”.
• Output using EXCEL software is given below:

Therefore, $P\left(X<66\right)=0.9772$.

Thus, the probability to reach at the airport in 66 minutes using route B is 0.9972.

The CV for route B is,

$\begin{array}{c}{\text{CV}}_B=\frac{3}{60}\times 100\%\\ =0.05\times 100\%\\ =5\%\end{array}$

Therefore, there is same chance to reach airport by route A and route B.

Now, the coefficient of variation for route B is less than route A.

Thus, the better route for John to reach at the airport in 66 minutes to pick up his spouse is route B.