## Friday, August 15, 2014

### Subaru versus Honda -- the tour guide's fuel consumption.

Question:  A tour guide drives his guest in an SUV.   The average tour is 50 miles of city driving.   The standard deviation of miles driven is 8 miles.  The tour guide can get a tour in the morning and a tour in the afternoon.   The probability of getting a tour in the morning is independent of getting a tour in the afternoon.   The probability of a morning tour is 0.5 and the probability of an afternoon tour is 0.7.

Each tour is exactly 150 miles driven in a city.   The tour can occur in a Honda CRV-V-EX with a fuel efficiency of 16mpg city or a Subaru XV Cross trek with a 21 mpg in the city.

What is the expected gas consumption for the tour operator with the Honda CRV and for the Subaru Cross Trek?

What is the variance of the daily gas consumption?

E(Gas) = E (Miles)/mpg

E(Gas Exp) = cpg * E(miles) / mpg

Here mpg is miles per gallon and cpg is cost per gallon.

Calculate expected miles.

E(Miles) = 0.5 *150 + 0.7 *150 = (75+90)=165.

Plug into the gas and consumed equations for the two cars.

Results summarized here.

 Tour Guide Fuel Consumption for Honda CRV and Subaru Cross Trek Hybrid Honda CRV Subaru Cross Trek City Fuel Efficiency 16 32 Expected Gas In Gallons 10.31 7.86 Expected Gas Expenditure  @ \$4.00 \$41.25 \$31.43

The variance of miles driven should be

150 x 0.5 x 0.5 + 150 x 0.75 x 0.25=65.625.

Look here for the formula of the variance of the Bernoulli distribution.

Using the properties of variance we can derive the variance of gas expenditures.

Var((ccg/mpg) x Miles)  = (cpg/mpg)2 x Var(miles)

 Variability of Tour Guide Gas Expenditures Honda CRV Subuaru Cross Trek cpg/mpg squared 0.0625 0.015625 Variance \$ Gas \$4.10 \$1.03 Standard Deviation\$ Gas \$2.03 \$1.01

Concluding Remark:  The variance of gas consumption and expenditures on gas consumption is much lower for the tour guide that uses the Subaru Cross Trek than the Honda CRV.

There is only one source of variance in gas consumption for the tour guide in this example  -- uncertainty the number of clients.  In the real world there is uncertainty about other variables including price of gas and miles driven per trip.   I am planning future posts that consider other factors affecting the variability of fuel consumed and expenditures on fuel.

#Subaru
#Honda
#fuel efficiency