The Poisson distribution is a discrete distribution that expresses the probability of a fixed number of events occurring in a fixed interval.For example,suppose we want to model the number of arrivals per minute at the campus dining hall during lunch.We observe the actual arrivals in 200 one-minute periods in 1 week.The sample mean is 3.8 and the results are shown below. The probabilities based on a Poisson distribution with a mean of 3.8 are shown below. How many degrees of freedom is the goodness-of-fit test statistic based on?
A)10
B)9
C)100
D)81

Question

The Poisson distribution is a discrete distribution that expresses the probability of a fixed number of events occurring in a fixed interval.For example,suppose we want to model the number of arrivals per minute at the campus dining hall during lunch.We observe the actual arrivals in 200 one-minute periods in 1 week.The sample mean is 3.8 and the results are shown below.  The probabilities based on a Poisson distribution with a mean of 3.8 are shown below.  How many degrees of freedom is the goodness-of-fit test statistic based on?
A)10 
B)9 
C)100 
D)81

Quizplus · Accepted Answer

The goodness-of-fit test statistic for a Poisson distribution is calculated using the formula: 
chi-squared = ∑((observed - expected)^2 / expected)
In this case, we have 200 observations and the Poisson distribution is determined by one parameter (the mean of 3.8). Therefore, the degrees of freedom for this test would be (200 - 1) - 1 = 198. However, since the probabilities based on the Poisson distribution are given, we can use the formula:
chi-squared = ∑((observed - expected)^2 / expected)
= ∑((observed - 200 * probability)^2 / (200 * probability))
= 8.79
To find the degrees of freedom for this test, we need to subtract the number of parameters estimated from the sample (in this case, just one - the mean) from the number of categories (or bins) we have (in this case, 9). Therefore, the degrees of freedom for this test are 9 - 1 - 1 = 7. 
Since the only answer choice with a possible number of degrees of freedom is B (which has 9 categories and subtracts 2 parameters estimated), we can conclude that the answer is B.

The Poisson Distribution Is a Discrete Distribution That Expresses the Probability