Disturbing news regarding Scottish police concerns the number of crashes involvi

Question

Disturbing news regarding Scottish police concerns the number of crashes involving vehicles on operational duties (BBC News, March 10, 2008). Statistics showed that Scottish forces' vehicles had been involved in traffic accidents at the rate of 1,000 per year. The statistics included vehicles involved in 999 calls (the equivalent of 911 in the U.S.) and pursuits. Fire service and ambulance vehicles were not included in the figures. Consider the expected number of traffic accidents per day involving vehicles on operational duties. Use Poisson function in Excel (POISSON.DIST), create probabilities for P(X= x) =? and P(X<=x)=? For the Poisson answers, construct the entire sample distribution: Based on the excel output, answer the following questions (Round your answers to 3 decimal places):

1. What is the expected number of traffic accidents per day involving vehicles on operational duties?_____________

2. What is the standard deviation of the Poisson probability distribution? _________

3. Calculate the Probability that exactly 3 traffic accidents per day involving vehicles on operational duties. __________________

4. Calculate the Probability that more than 3 traffic accidents per day involving vehicles on operational duties. __________________

5. Calculate the Probability that at most 5 traffic accidents per day involving vehicles on operational duties. __________________

Part 2 Poission average 1000 per year Average per day 2.73973 2.74 0 0.064570.0645703 1 0176920.2414931 2 0.24238 0.4838773 3 0.221380.7052548 0.15164 0.8568984 5 0.08310 0.9399991 0.03795 0.9779484 7 0.01485 0.9928029 8 0005090.9978905 0.00155 0.9994394 10 0.00042 0.9998638 11 0.00011 0.9999696 122.4E-05 0.9999937 135.1E-06 0.9999988 1.0E-06 0.9999998 14 1.8E-07 3.1E-08 16

Explanation / Answer

Ans:

a)mean=1000 per year=1000/365 per day=2.7397 per day

b)standard deviation=sqrt(2.7397)=1.6552

c)P(x=3)=POISSON(3,2.7397,FALSE)=0.22137

d)P(x>3)=1-P(x<=3)=1-POISSON(3,2.7397,TRUE)=1-0.705321=0.29468

e)P(x<=5)=POISSON(5,2.7397,TRUE)=0.940024

x P(X=x) P(X<=x) 0 0.064590 0.064590 1 0.176956 0.241546 2 0.242404 0.483950 3 0.221371 0.705321 4 0.151623 0.856944 5 0.083080 0.940024 6 0.037936 0.977960 7 0.014848 0.992807 8 0.005085 0.997892 9 0.001548 0.999440 10 0.000424 0.999864 11 0.000106 0.999970 12 0.000024 0.999994 13 0.000005 0.999999 14 0.000001 1.000000 15 0.000000 1.000000 16 0.000000 1.000000 17 0.000000 1.000000

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