Given a table of flights with their origin, destination, and flight time, we would like to estimate the length of a proposed flight from Bratislava, Slovakia to Amsterdam, the Netherlands:

Analysis:

We can reason that the flight duration time consists of two times - the first is the time to take off and the landing time; the second is the time that the airplane moves at a certain speed in the air. The first time is some constant. The second time depends linearly on the speed of the plane, which we assume is similar across all the flights in the table. Therefore, the flight time can be expressed using a linear formula in terms of the flight distance.

Analysis using R:

Input:

source_code/6/flight_time.r 
flights = data.frame( 
    distance = c(365,1462,1285,1096,517,1686,932,1160), 
    time = c(1.167,2.333,2.250,2.083,2.250,2.833,1.917,2.167) 
) 
model = lm(time ~ distance, data = flights) print(model) 

Output:

$ Rscript flight_time.r
Call:
lm(formula = time ~ distance, data = flights)
Coefficients: (Intercept)     distance
                1.2335890    0.0008387  

According to the linear regression, the time to take off and the landing time for an average flight is about 1.2335890 hours. Then to travel 1 km with the plane takes 0.0008387 hours; in other words, the speed of an airplane is 1192 km per hour. The actual usual speed of an aeroplane for short-distance flights like the ones in the table is about 850 km per hour. This leaves room for improvement in our estimation (refer to exercise 6.3).

Therefore, we derived the following formula:

flight_time=0.0008387*distance+1.2335890 hours

Using it, we estimate that the flight from Bratislava to Amsterdam, with the distance 978 km, would take about 0.0008387*978+1.2335890=2.0538376 hours or about 2 hours and 3 minutes, which is a little longer than from Vienna to Amsterdam (1h 55m) and a little shorter than from Budapest to Amsterdam (2h 10m).