Question

Taxi! In $2016$, taxicabs in Los Angeles charged an initial fee of $\backslash (2.85$ plus $\backslash )2.70$ per mile. In equation form, Fare$=2.85+2.7\left(miles\right)$. At the end of a

month, a businessman collects all his taxicab receipts and calculates some numerical

summaries. The mean fare he paid was $\backslash (15.45$with a standard deviation of $\backslash )10.20$What are the mean and standard deviation of the lengths of his cab rides in miles?

Short answer

Mean in miles, ${\mu }_{mi.}=4.6667miles$

Standard deviation in miles,${\sigma }_{mi.}=3.7778miles$

Step-by-step solution

Given information

$Fare=2.85+2.7\left(miles\right)$

Mean, ${\mu }_{\$}=\$15.45$

Standard deviation,${SD}_{\$}=\$10.20$

Calculation

To get the number of miles from the fare, we must first reduce it by $2.85$ and then divide the result by $2.7$

Since

$Fare=2.85+2.7\left(miles\right)$

That becomes

$miles=\frac{Fare-2.85}{2.07}$

Mean :

Because the mean represents the measurement's centre.

When each data value is reduced by $2.85$ the measurement's centre is similarly reduced by $2.85$

Similarly,

When all data values are divided by $2.7$, the measurement's centre is likewise divided by $2.7$

Then

The new standard deviation

${\sigma }_{mi.}=\frac{{\sigma }_{\$}}{2.7}=\frac{10.20}{2.7}\approx 3.7778miles$

Thus,

The standard deviation of the length of the cab ride in miles is $3.7778$ miles.