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 2.7 (type an integer or a decimal. do not round.) the standard deviation is 1.64 (round to two decimal places as needed.) interpret the results. choose the correct answer below. a. the number of bankruptcies is always the standard deviation number of bankruptcies b. the number of bankruptcies usually differs from the mean by no more than the standard deviation number of bankruptcies c. the number of bankruptcies differs from the standard deviation by no more than the mean number of bankruptcies. d. the number of bankruptcies always differs from the mean by no more than the standard deviation number of bankruptcies.

Explanation:

Step1: Recall the property of Poisson distribution variance

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

Step2: Calculate the standard deviation

The formula for the standard deviation \(\sigma=\sqrt{\sigma^{2}}\). Substitute \(\sigma^{2} = 2.7\) into the formula, \(\sigma=\sqrt{2.7}\approx1.64\) (rounded to two decimal places).

Step3: Interpret the meaning of standard deviation

The standard deviation measures the spread of the data. In a Poisson - distributed data set (or in general for many distributions), the data (the number of bankruptcies in this case) usually differs from the mean by no more than the standard deviation number of bankruptcies.

Answer:

a) Variance: \(2.7\), Standard deviation: \(1.64\)
b) Interpretation: B. The number of bankruptcies usually differs from the mean by no more than the standard deviation number of bankruptcies.