Chapter 6.2, Problem 60E

(a)

To determine

Meaning for B and W assumed as independent random variables in context.

Answer to Problem 60E

Independent random variables B and W does not affect each other’s commute’s time.

Explanation of Solution

Given information:

B and W are independent random variables.

For B:

Mean,

  ${\mu }_{\text{B}}=12\text{minutes}$

Standard deviation,

  ${\sigma }_{\text{B}}=4\text{minutes}$

For W:

Mean,

  ${\mu }_{\text{W}}=16\text{minutes}$

Standard deviation,

  ${\sigma }_{\text{W}}=1\text{minute}$

B: commute time by bus

W: commute time when walking

We know that

B and W are independent random variables.

This implies

In any way, the commute time by bus B does not affect the commute time when walking W.

Similarly,

In any way, the commute time when walking W does not affect the commute time by bus B.

(b)

To determine

Interpretation and calculation of standard deviation of the difference in time it would take Sule to get to work on a randomly selected day.

Answer to Problem 60E

On an average, the difference between the commute time by bus and the commute time when walking deviates by approx. 4.1231 minutes from the mean difference of -4 minutes.

Explanation of Solution

Given information:

B and W are independent random variables.

For B:

Mean,

  ${\mu }_{\text{B}}=12\text{minutes}$

Standard deviation,

  ${\sigma }_{\text{B}}=4\text{minutes}$

For W:

Mean,

  ${\mu }_{\text{W}}=16\text{minutes}$

Standard deviation,

  ${\sigma }_{\text{W}}=1\text{minute}$

B: commute time by bus

W: commute time when walking

Now,

Mean difference of the mean of commute times for both B and W:

  ${\mu }_D={\mu }_{\text{B}-\text{W}}={\mu }_{\text{B}}-{\mu }_{\text{W}}=12-16=-4\text{minutes}$

When the random variables are independent, the variance of the difference is equal to the sum of their variances.

  ${\sigma }_D^2={\sigma }_{\text{B}-\text{W}}^2={\sigma }_{\text{B}}^2+{\sigma }_{\text{W}}^2={\left(4\right)}^2+{\left(1\right)}^2=16+1=17\text{minutes}$

We also know that

The standard deviation is the square root of the variance:

  ${\sigma }_{\text{B}-\text{W}}=\sqrt{{\sigma }_{\text{B}-\text{W}}^2}=\sqrt{17}\approx 4.1231\text{minutes}$

On an average, the difference between the commute time by bus and the commute time when walking deviates by approx. 4.1231 minutes from the mean difference of -4 minutes.

(c)

To determine

Whether it is possible to find the probability for Sule takes longer to get to work on the bus than if he walks on a randomly selected day.

Answer to Problem 60E

No, because the distributions of B and W are not known.

Explanation of Solution

Given information:

B and W are independent random variables.

For B:

Mean,

  ${\mu }_{\text{B}}=12\text{minutes}$

Standard deviation,

  ${\sigma }_{\text{B}}=4\text{minutes}$

For W:

Mean,

  ${\mu }_{\text{W}}=16\text{minutes}$

Standard deviation,

  ${\sigma }_{\text{W}}=1\text{minute}$

B: commute time by bus

W: commute time when walking

From Part (b),

We came to know that

The difference B-W had mean of -4 minutes and standard deviation of approx. 4.1231 minutes.

Since the distribution of B or the distribution W is not known.

Thus,

The distribution of the difference B-W cannot be determined.

Hence,

It is impossible to determine the probability that the commute time by bus exceeds the commute time when walking.