Chapter 18.2, Problem 43E

Time of Traffic Fatality In Exercise 51 of the previous section, the probability density function for the number of fatal traffic accidents was found to be

$S\left(t\right)=\frac{1}{91,762}\left(-2.416t^3+90.91t^2-846.8t+4880\right)$

where t is the number of hours since midnight on [0, 24]. Calculate the expected time of day at which a fatal accident will occur. Source: The National Highway Traffic Safety Administration.

51.    Time of Traffic Fatality The National Highway Traffic Safety Administration records the time of day of fatal crashes. The table on the next page gives the time of day (in hours since midnight) and the frequency of fatal crashes in 2012. Source: The National Highway Traffic Safety Administration.

(a)    Plot the data. What type of function appears to best match these data?

(b)    Use the regression feature on your graphing calculator to find a cubic equation that models the time of day, t, and the number of traffic fatalities, T(t). Use the midpoint value to estimate the time in each interval. Graph the function with the plot of the data. Does the graph fit the data?

Time of Day	Midpoint of Interval (hours)	Frequency
0–3	1.5	3817
3–6	4.5	2560
6–9	7.5	2950
9–12	10.5	2982
12–15	13.5	3973
15–18	16.5	4964
18–21	19.5	5003
21–24	22.5	4310
Total		30,559

(c)    By finding an appropriate constant k, find a function S(t) – kT(t) that is a probability density function describing the probability of a traffic fatality at a particular time of day.

(d)    For a randomly chosen traffic fatality, find the probabilities that the accident occurred between 12 A.M. and 2 A.M. (t = 0 to t = 2) and between 4 P.M. and 5:30 P.M. (t = 16 to t = 17.5).