Poisson distrubution / exponential random variable

An employee at a gym recorded the number of people coming in to work out or attend physical therapy during each half hour that he worked.

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

B. His recorded data is: 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

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?

# of people Frequency

0 3

1 2

2 7

3 7

4 5

5 4

__6 2__

30

C. The time between people arriving is:

Time Frequency

1 7

2 6

3 7

4 18

5 13

6 13

7 7

8 9

9 6

__10 4__

90

If X is an exponential random variable with mean $\displaystyle \theta=5 $, compare the probability P(X>2) with the proportion of times that are greater than 2 from the data provided.

Thank you so much for any help you can give me!