### Current data

The current information which I have are fuel consumption of last year. I have taken a set of those data from March and April of 2014.library(MASS)

r1 <- read.table(text='l km

28.25 710.6

22.93 690.4

28.51 760.5

23.22 697.9

31.52 871.2

24.68 689.6

30.85 826.9

23.04 699

29.96 845.3

30.16 894.7

25.71 696

23.6 669.8

28.57 739

27.23 727.4

18.31 499.9

24.28 689.5',header=TRUE)

r1$usage=100*r1$l/r1$km

plot(density(r1$usage),

main='Observed normal diesel usage',

xlab='l/100 km')

The data are from a distribution with a mean around 3.6 l/100 km.

fitdistr(r1$usage,'normal')

mean sd

3.59517971 0.19314598

(0.04828649) (0.03414371)

### Approach

Analysis will be a hypothesis test and an estimate of premium diesel usage.

The assumptions which I will make are similar driving patterns and weather as last year. I think that should be possible, given my driving style. A cross check may be made, especially regarding obtaining similar speed. Data with serious traffic jams may be discarded in the analysis.

A check for outliers is not planned. However, obviously faulty data will be corrected or removed from the data. No intermediate analysis is planned, unless data seems to be pointing a marked increase of fuel usage.

The assumptions which I will make are similar driving patterns and weather as last year. I think that should be possible, given my driving style. A cross check may be made, especially regarding obtaining similar speed. Data with serious traffic jams may be discarded in the analysis.

A check for outliers is not planned. However, obviously faulty data will be corrected or removed from the data. No intermediate analysis is planned, unless data seems to be pointing a marked increase of fuel usage.

### Power for hypothesis test

The advice price levels of premium and standard diesel are 1.433 and 1.363 Euro/liter according to the internet. This is about 5% price increase. It should be noted that prices at the pump vary wildly from these values, especially non-brand non-manned fuel stations may be significantly cheaper. Last year's data was from such non brand fuel. Competition can force the price of both standard and premium fuel down a bit. I will take the 5% price increase as target for finding value for premium diesel. Given significance level of 10% and power of 90%, I come at 17 samples for each group. This means I will have to take a bit more data from last year, which is not a problem. The choice of alpha and beta reflect that I find both kind of errors equally bad.

power.t.test(delta=3.6*.05,

sd=0.2,

sig.level=.1,

power=.9,

alternative='one.sided')

Two-sample t test power calculation

n = 16.66118

delta = 0.18

sd = 0.2

sig.level = 0.1

power = 0.9

alternative = one.sided

NOTE: n is number in *each* group

### Estimating usage of premium diesel

Besides a significance test, I desire an estimate of usage. This manner I can extrapolate the data to other scenarios. I will use a Bayesian analysis to obtain these estimates. The prior to be used is a mixture of three believes. Either it does not make a difference, or there is indeed a 5% gain to be made or something else entirely. This latter is an uninformed prior between 3 and 4 l/km. The combined density is plotted below.

usage <- seq(3.2,3.8,.01)

dens <- (dnorm(usage,3.6,.05)+

dnorm(usage,3.6/1.05,.05)+

dnorm(usage,(3.6+3.6/1.05)/2,.15))/3

plot(x=usage,y=dens,type='l',

ylim=c(0,4),

ylab='density',

xlab='l/100 km',

main='prior')

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