The National Highway Traffic Safety Administration records the time of day of fatal crashes. The following table gives the time of day (in hours since midnight) and the frequency of fatal crashes. 

a. Plot the data. What type of function appears to best match this 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?

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 am and 2 am (t = 0 to t = 2) and between 4 pm and 5:30 pm (t = 16 to t = 17.5).

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Time of Day 0-3 3-6 6-9 9-12 12-15 15-18 18-21 21-24 Total Midpoint of Interval (hours) 1.5 4.5 7.5 10.5 13.5 16.5 19.5 22.5 Frequency 4486 2774 3236 3285 4356 5325 5342 4952 33,756