Question

Let X equal the number of telephone calls per hour that are received by 911 between midnight and noon. For a two-day period, researchers recorded the number of calls received in each hour. Those values were: 0, 1, 1, 1, 0, 1, 2, 1, 4, 1, 2, 3, 0, 3, 0, 1, 0, 1, 1, 2, 3, 0, 2, 2.

This problem is about deciding if it looks like the Poisson distribution might be an appropriate model for X.

(a) First, recall that, for a Poisson distribution, the mean and the variance are the same. Calculate the mean and variance for this dataset. Are they approximately equal? What does this tell you about the plausibility modeling this data as coming from a Poisson distribution?

(b) Next, compare the numbers of calls that arrived with what you would expect from a Poisson distribution. Specifically, for each value 0, 1, 2, 4, and “5 or more”, calculate the probability that a Poisson distribution will yield that value. Use the Poisson distribution with µ equal to the mean of your dataset. Then compare this to the proportion of hours in the dataset that had that many calls. Please give me this comparison in two forms: A table and a graph. Use some of your own judgement on this one. In the end, comment on whether you believe the data follows a Poisson distribution.

Answer 1

a) Mean of the given data is 1.33333 and Variance is 1.275362. hence the data may have come from poisson distribution. They are approximately same. The values obtained are results from R output.

b)