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B and W are two unrelated random variables.

To get the B:

Mean,

${\mathrm{μ}}_{\mathrm{B}}=12$minutes

Standard deviation,

${\mathrm{σ}}_{\mathrm{B}}=4$minutes

For W:

Mean,

${\mathrm{μ}}_{\mathrm{W}}=16$minutes

Standard deviation,

${\mathrm{σ}}_{\mathrm{w}}=1$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.

B and W are two unrelated random variables.

For B:

Mean,

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

Standard deviation,

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

For W:

Mean,

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

Standard deviation,

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

B: commute time by bus

W: commute time when walking

Now,

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

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

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

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

We also know that

The square root of the variance is the standard deviation:

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

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

B and W are two unrelated random variables.

For B:

Mean,

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

Standard deviation,

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

For W:

Mean,

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

Standard deviation,

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

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.