A mathematics student worked part time in the training room

A mathematics student worked part time in the training room in the physical education building.  This sudent recorded the number of students coming in to be taped or for treatment of other injuries during each half hour she worked.

(a)  Explain how a Poisson distribution and an exponential distribution could be involved in her recording of data.  Be very specific in how each distribution could apply.

(b)  Her recorded data are: 3 5 2 3 4 3 6 4 3 0 1 2 3 2 0 6 2 4 2 3 4 2 4 1 5 5 0 2 3 5

Find (i)  the sample mean  (ii) the sample variance  (iii) the sample standard deviation

(c)  Construct a frequency table comparing the data to an appropriate Poisson distribution. Does the comparison support the conjecture that the mean number of arrivals is approximately Poisson?

(d)  The time between students arriving for taping or treatment is :  6 9 6 4 4 4 4 4 10 9 6 5 18 8 7 7 3 10 8 4 4 4 2 3 4 5 7 8 4  5 7 9 5 7 3 5 5 4 5 6 2 1 1 8 9 7 7 4 5 10 5 6 5 4 4 8 9 5 9 5 8 4 1 6 6 6 4 2 6 6 6 3 1 2 3 8 1 2 3 4 6 8 1 3 10 5 2 6 4

(e)  If X is an exponential random variable with mean \"theta=5,\" compare the probability P( X > 2)with the proportion of times that are greater than 2 from the data in (d)

Solution

The units for Poisson are People (or arrivals), the units for Exponential are minutes.

Average customers per hour is 5 means Poisson with mean 5.

The same if converted to waiting time in minutes then waiting time varies from 0 to 20 here exponential distribution is used.

Poisson parameter is 6 per hour means exp. parameter =10 min

for 10 exp -6 min and so on.

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Total Numbers

30

Mean (Average)

2.96667

Population Standard deviation

1.64283

Variance(Population Standard deviation)

2.69889

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This can be fitted into POisson as mean ~ Variance.

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d) Freq distribution for the data given :

(X>2) as per table = 91-16 = 75

Prob = 75/91 = 0.824

As per exponential distribution

P(X>2) = 0.2681

No of people =