7.25 Train Distance & Travel Time
The Coast Starlight, Part II. Exercise 7.13 introduces data on the Coast Starlight Amtrak train that runs from Seattle to Los Angeles. The mean travel time from one stop to the next on the Coast Starlight is 129 mins, with a standard deviation of 113 minutes. The mean distance traveled from one stop to the next is 108 miles with a standard deviation of 99 miles. The correlation between travel time and distance is 0.636.
Point-slope equation, will use calculations below to answer parts a-f.
\(b_1\) = \(\frac{113}{99}\)(.636) \(b_1\) = .725
y-\(*y*_0\) = \(*b*_1(x - x_0)\) y - 129 = .725(x- 108) y = 50.7 + .725X
r = .636 \(r^2\) = .404
Write the equation of the regression line for predicting travel time.
y = 50.7 + .725X
Interpret the slope and the intercept in this context.
The intercept says that travel time will be 50.7, when there is no distance traveled - which probably has no pratical value or it could mean that other factors such as congestion or weather determine up to 50.7 minutes of travel time. The slope tells us that for one mile will be traveled, it will take per .725 of a minute or 44 seconds.
Calculate R2 of the regression line for predicting travel time from distance traveled for the Coast Starlight, and interpret R2 in the context of the application.
\(r^2\) = .404, this says that 40% of the variation travel can be exlplained by distance, there are other factors outside of this model such as congestion, weather that might also i infleunce travel time. This model is very weak in explaining the variation in y.
The distance between Santa Barbara and Los Angeles is 103 miles. Use the model to estimate the time it takes for the Starlight to travel between these two cities.
The distance from Santa Barbara to Los Angeles is 103 miles. This is the expected time for our arrival: 50.7 + .725(103) = 125.38
It actually takes the Coast Starlight about 168 mins to travel from Santa Barbara to Los Angeles. Calculate the residual and explain the meaning of this residual value.
The observed travel time is 168. The residual is as follows, 168 - 125.38 = 42.62, this represents error - the difference between the expected and observed value, it actually took 42.62 more minutes to travel from Santa Barbara to LA than expected.
Suppose Amtrak is considering adding a stop to the Coast Starlight 500 miles away from Los Angeles. Would it be appropriate to use this linear model to predict the travel time from Los Angeles to this point?
Given that mean distance is 108 miles, and SD = 99, going three standard deviations from the mean results in 397 miles. I don’t 500 miles is too far beyond this mark. However given the large standard deviations and low \(R^2\) I don’t think this model is reliable at any distance. It’s likely the slope for distance doesn’t have the t-stat and p-values to be statisically valid anyway.