Question

Commuting to work Refer to Exercise 52 .

a. Assume that B and $\mathrm{W}$are independent random variables. Explain what this means in context.

b. Calculate and interpret the standard deviation of the difference $D$(Bus - Walk) in the time it would take Sulé to get to work on a randomly selected day.

c. From the information given, can you find the probability that it will take Sulé longer to get to work on the bus than if he walks on a randomly selected day? Explain why or why not.

Short answer

1. The commute times of independent random variables B and W are unaffected by each other.
2. The average difference between commuting by bus and commuting by foot deviates by approximately $4.1231$minutes from the mean difference of $-4$minutes.
3. No, because the B and W distributions are unknown.

Step-by-step solution

Part (a) Step 1: Given information

B and W are two unrelated random variables.

Part (a) Step 2: Calculation

To get the B:

Mean,

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

Standard deviation,

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

For W:

Mean,

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

Standard deviation,

${\sigma }_{\mathrm{w}}=1$minute

B: bus travel time

W: walking commute time

We are aware that

B and W are two unrelated random variables.

This means

In any case, the commute time by bus B has no bearing on the journey time by walking W.

Similarly,

In any case, the commute time by walking W has no bearing on the commuting time by bus B.

Part (b) Step 1: Given information

B and W are two unrelated random variables.

Part (b) Step 2:  Calculation

For B:

Mean,

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

Standard deviation,

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

For W:

Mean,

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

Standard deviation,

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

B: commute time by bus

W: commute time when walking

Now,

The mean difference in travel times for B and W is:

$\begin{array}{rcl}{\mu }_D& =& {\mu }_{\mathrm{B}-\mathrm{W}}={\mu }_{\mathrm{B}}-{\mu }_{\mathrm{w}}\\ & =& 12-16\\ & =& -4\text{minutes}\end{array}$

The variance of the difference is equal to the sum of the variances of the random variables when they are independent.

$\begin{array}{rcl}{\sigma }_D^2& =& {\sigma }_{\mathrm{B}-\mathrm{w}}^2={\sigma }_{\mathrm{B}}^2+{\sigma }_{\mathrm{W}}^2\\ & =& (4{)}^2+(1{)}^2\\ & =& 16+1\\ & =& 17\text{minutes}\end{array}$

We also know that

The square root of the variance is the standard deviation:

$\begin{array}{rcl}{\sigma }_{\mathrm{B}-\mathrm{w}}& =& \sqrt{{\sigma }_{\mathrm{B}-\mathrm{w}}^2}\\ & =& \sqrt{17}\\ & \approx & 4.1231\text{minutes}\end{array}$

The average difference between commuting by bus and commuting by foot deviates by approximately 4.1231 minutes from the mean difference of -4 minutes.

Part(c) Step 1: Given information

B and W are two unrelated random variables.

Part(c) Step 2: Calculation

For B:

Mean,

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

Standard deviation,

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

For W:

Mean,

${\mu }_{\mathrm{w}}=16\text{minutes}$

Standard deviation,

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

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.

Because the distribution of B or the distribution of W are unknown.

Thus,

The distribution of the difference B-W is unknown.

Hence,

It is hard to predict whether the commute time by bus will be longer than the travel time by walking.