Question

the mean number of bankruptcies filed per hour by businesses in a country one year was about 2.7.
(a) find the variance and standard deviation using the fact that the variance of a poisson distribution is ( sigma^{2}=mu ). interpret the results
(b) identify the least number of bankruptcies during an hour that you would consider unusual.
click the icon to view the table of poisson probabilities
(a) the variance is
(type an integer or a decimal. do not round.)

Explanation:

Step1: Calculate the variance

Given that for a Poisson distribution, the variance \(\sigma^{2}=\mu\). Here, \(\mu = 2.7\), so \(\sigma^{2}=2.7\)

Step2: Calculate the standard deviation

The formula for the standard deviation \(\sigma\) is \(\sigma=\sqrt{\sigma^{2}}\). Substituting \(\sigma^{2}=2.7\), we get \(\sigma=\sqrt{2.7}\approx1.64\)

Interpretation: The variance of \(2.7\) represents the spread or dispersion of the number of bankruptcies around the mean. A higher variance (in relation to the mean) indicates more variability in the number of bankruptcies per hour. The standard deviation of approximately \(1.64\) is the square - root of the variance and also measures the spread. Values within about \(2\) standard deviations of the mean (\(\mu\pm2\sigma\)) are considered usual.

Answer:

(a) The variance is \(2.7\)