Is my 07:40 bus late because of the rain?

My school gives a late mark to anyone who arrives after registration at 08:35, and three late marks in a term mean an after-school detention. I take the 07:40 bus, which is timetabled to reach the stop outside school at 08:18, and from there it is a ten-minute walk to my form room, so I need to be off the bus by 08:25. Last year I got four late marks, all — as far as I remember — on rainy days. My mother's view is that “the bus is only ever late when it rains” and that I should take the earlier 07:25 bus on wet mornings; I would rather have fifteen more minutes in bed.

Aim. Using a log of 64 journeys, I will (1) describe how reliable the 07:40 bus is and decide whether a normal distribution is a reasonable model for its arrival time, (2) test whether wet weather and late arrival are associated, (3) find out whether rain makes the journey itself slower or only delays the bus leaving, and (4) use the models to decide which bus I should take on wet and dry days.

The decision in part (4) is the reason for the whole exploration, so I will judge every result by whether it helps me make it.

From the second week of September to the middle of December I recorded every school-day journey on the 07:40 bus: 64 days in all (I was ill on two days and one day the bus did not run). For each journey I noted:

\(D\) = departure delay: minutes between 07:40 and the time the bus actually left my stop, from the operator's live-tracking app (to the nearest minute);
\(J\) = journey time: minutes from leaving my stop to arriving at the school stop, from the same app;
\(A\) = arrival time at the school stop, in minutes after 08:00, so \(A = -20 + D + J\);
weather: wet if it was raining at my stop at 07:30, otherwise dry.

I chose to record the weather myself at 07:30 rather than use a daily rainfall total, because a day with rain in the afternoon is a dry morning for my bus. I defined late as \(A \ge 26\) (after 08:25), which in a continuous model means \(A > 25.5\), because the app rounds to the nearest minute. Taking 25.5 rather than 25 matters: with a standard deviation of about 3 minutes, half a minute changes a probability of being late by several percentage points.

Before this, over the last six weeks of the summer term, I had kept a pilot log of 30 journeys on the same bus, mainly to learn how the app works (Appendix D). I did not record the weather in it, so I use it only once: in Section 4, as a normal model fixed before the main data were collected.

Data note: the journey log in this exemplar was simulated by IB Math Revision to behave like a real one. A student must collect and use their own data.

Two limitations were clear before I started. My observations are 64 consecutive school days in one autumn, so they may not represent winter snow or summer traffic, and the days are not truly independent: one week had roadworks on the main road, which could make several days in a row slow together. I kept a note of the roadworks days (Appendix C) so that I could look at them separately if they stood out.

A \(\chi^2\) test of independence. My first plan was a 2 × 2 table of wet/dry against late/not late. But only 10 days were late, and the expected frequency for “wet and late” would have been \(\frac{24 \times 10}{64} = 3.75\), below 5, so the test would not have been valid. Instead I used three arrival categories whose boundaries mean something to me: by 08:18 (on time), 08:19–08:23 (late bus, but I still make registration comfortably), and 08:24 or later (I have to run, or I am late).

\(H_0\): arrival category is independent of the weather. \(H_1\): arrival category is not independent of the weather. Significance level 5%.

Each expected frequency is \(\frac{\text{row total} \times \text{column total}}{64}\); for example the expected number of wet days with arrival by 08:18 is \(\frac{24 \times 17}{64} = 6.375\), and its contribution to the statistic is \(\frac{(0 - 6.375)^2}{6.375} = 6.375\). All six expected frequencies are at least 5. Adding all six contributions gives \(\chi^2_{\text{calc}} = 21.22\) with \((2-1)(3-1) = 2\) degrees of freedom, so \(p \approx 2.5 \times 10^{-5}\) (critical value 5.991).

Since \(p < 0.05\) I reject \(H_0\): there is very strong evidence that arrival time and weather are associated. The largest contributions come from the “08:24 or later” column (6.88 for wet days) and from the fact that the bus never arrived on time on a wet day in my log (0 observed against 6.38 expected).

Does rain slow the journey itself? The \(\chi^2\) test says nothing about why. To see whether the journey is slower, and not only the departure, I compared mean journey times with a one-tailed two-sample \(t\)-test. \(H_0\): \(\mu_{\text{wet}} = \mu_{\text{dry}}\); \(H_1\): \(\mu_{\text{wet}} > \mu_{\text{dry}}\). The \(t\)-test assumes that both samples come from normal distributions; the journey times in each group are roughly symmetric with no outliers, and the standard deviations (2.25 and 1.99 minutes) are similar, so I used the pooled version:

\[ s_p = \sqrt{\frac{(n_w - 1)s_w^2 + (n_d - 1)s_d^2}{n_w + n_d - 2}} = \sqrt{\frac{23 \times 2.245^2 + 39 \times 1.994^2}{62}} \approx 2.091, \]

\[ t = \frac{\bar x_w - \bar x_d}{s_p\sqrt{\frac{1}{n_w} + \frac{1}{n_d}}} = \frac{41.542 - 37.850}{2.091 \times \sqrt{\frac{1}{24} + \frac{1}{40}}} \approx 6.84, \]

with 62 degrees of freedom, giving \(p \approx 2 \times 10^{-9}\) (my GDC gave the same value). This is far below 0.05, so there is strong evidence that the mean journey time is longer on wet days — by about 3.7 minutes in my sample. So rain does not only make the bus leave late: it makes the whole journey slower, which is what I would expect with heavier traffic and passengers who take longer to board with umbrellas.

The 07:40 bus is reliable on dry mornings (median arrival 08:19, one minute after the timetable) and unreliable on wet ones (median 08:24). A \(\chi^2\) test of independence showed very strong evidence that arrival time is associated with the weather (\(p \approx 2.5 \times 10^{-5}\)), and a \(t\)-test showed that the journey itself is slower on wet days by about 3.7 minutes, not only the departure. The apparent link between late departures and slow journeys is partly explained by the weather: within wet days and within dry days it is weaker. Normal models give a probability of being late of about 0.31 on a wet day and at least 0.02 on a dry day, so my plan is to take the 07:25 bus on wet mornings, which should reduce my late marks from about 8 to about 1 a term. My mother was mostly right, though not entirely: the bus is late on dry days occasionally, too.

The main limitations are in the data rather than the methods. My log covers one autumn, so it says nothing about winter ice or the summer term; “wet” is a yes/no judgement at one moment, which cannot distinguish drizzle from a downpour; and consecutive days are not independent, so the tests may be slightly over-confident. A rainfall measurement in millimetres from a weather station near the route would let me model the effect of rain as a continuous variable, and a second term's data would show whether the conclusions hold in winter.

Doing this changed how I see the timetable: it is not a promise but the centre of a distribution, and the question worth asking is not “is the bus late?” but “how likely is it to be late enough to matter to me?”