Question

# You are given the frequency distributions of the monthly totals of

industrial accidents for a period of 100

months at each of four sites. In each case, k

denotes the number of accidents, while nik denotes the frequency of months which

recorded k accidents for Site i, i=1,2,3,4. The data is given in indusacc.csv.

(a) By a suitable graphical method, determine whether the data in Site 1 conforms

to

(i) a Poisson distribution;

(ii) a negative binomial distribution.

Discuss which of the two probability distributions is more appropriate for Site 1.

(4 points)

(b) Compute the sample mean and sample variance for the number of accidents at

Site 1. Discuss your ndings in relation to your answer in part (a). (2 points)

(c) Repeat parts (a.i) and (b) for Site 4. (6 points)

(d) Repeat part (a.i) only for Sites 2 and 3, and also compute the sample mean and

sample variance for Sites 2 and 3. If we were to consider the negative binomial

distribution instead, determine the method of moments estimate of r for each of

Sites 2 and 3. What is the nature of this parameter r in terms of the dispersion of

the data? Based on the estimate you obtain, is the negative binomial distribution

appropriate for these two sites? Explain. (8 points)

(e) Use the technique discussed in the Lecture Notes (Note 7) to determine whether

Site 4 has too many zeros" for the Poisson distribution. (2 points)

(f) Under a Poisson assumption (with parameter estimated from the graphical

method in part (c)), compute the expected frequencies of accidents for Site 4.

Similarly under a negative binomial assumption, compute the expected frequen-

cies of accidents for Site 4. Comment on how the expected frequencies from each

distribution compare with the observed frequencies. (3 points)

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- Study Material used: Basic Statistics for Business & Economics by Lind Douglas A., Marchal William G. & Wathen, Samuel A. (2019) 9th Edition, McGraw Hill.