A Simple Simulation of Waiting for a Bus
The Seattle northbound C line stops at Third Avenue and Pike Street. Many of the buses are scheduled to run 12 minutes apart so for simplicity, we will assume that they are all scheduled to run exactly 12 minutes apart.
How long would you expect to wait for the bus if you arrive at the stop at a random time in a 12-minute period? How might the actual wait times differ from the expected time?
You might be able to answer these questions analytically if you have studied advanced probability. But, you can quickly get approximate answers with a simple simulation.
Suppose:
You arrive at random time in a 12-minute interval.
Suppose that times are measured to the nearest 1/100 of a minute.
1. Build a box model that represents this process, and produce 100,000 draws from the box with replacement. The box and the draws from the box should both be saved as vectors. Set the seed before taking your draws using
Hints: a) You might find the function useful here to create the tickets that go
in the box and b) there should be 1200 or 1201 tickets in the box.
2. Create a histogram of the approximate probability distribution of arrival time. What is the mean arrival time? The SD of arrival time?
3. Convert the simulated arrival times into wait times (recall the bus comes every 12 minutes). Repeat the computations from above to find the average wait time, standard deviation of wait time, and approximate probability distribution of the wait time.
4. Are you surprised by the size of the standard deviation? Why do you think this probability distribution is called “uniform”?
The blogger said that his experience did not match this simulation so he went and got some data about the arrival times of the bus.
Understanding the Data for Real Buses
The blogger acquired data from the Washington State Transportation Center for this bus line2 and two others. The data are for March 26 through May 27, 2016.
We have simplified these data and provided you with only information about how late each bus was in this 2 month time period.
5. Visualize and describe the empirical distribution of the lateness of the northbound C buses stopping at Third and Pike in this two-month period. Is it unimodal? Is it skewed left, skewed right, or symmetric? Are there any surprising values?
6. Summarize the typical amount of time the bus is late and compute a measure of spread to accompany your typical time.
7. How might the distribution of the actual arrival times at the bus stop impact the waiting time experience by the blogger?
To answer this question, we will carry out one more simulation. But before you do, take a stab at what you think this distribution might look like.
Will if be flat like the distribution from the first part?
Will it be simiilar to the histogram of the lateness of the bus, but shifted? Do you think the SD will be close to the SD from first simulation, the observed SD of bus lateness, or some other larger/smaller value?
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