Question

use the fact that the variance of a poisson distribution is $\sigma^2 = \mu$.
the mean number of bankruptcies filed per hour by businesses in a country was about seven. (a) find the variance and the standard deviation. interpret the results. (b) find the probability that at most five businesses will file bankruptcy in any given hour.

(a) the variance is 7.000
(round to three decimal places as needed.)

the standard deviation is 2.6
(round to one decimal place as needed.)

choose the correct answer below.

a. the number of bankruptcies differs from the standard deviation by no more than the mean number of bankruptcies.

b. the number of bankruptcies always differs from the mean by no more than the standard deviation number of bankruptcies.

c. the number of bankruptcies is always the standard deviation number of bankruptcies.

d. the number of bankruptcies usually differs from the mean by no more than the standard deviation number of bankruptcies.

Explanation:

To determine the correct interpretation of the standard deviation in the context of a Poisson distribution (where mean \(\mu = 7\), variance \(\sigma^2 = 7\), and standard deviation \(\sigma \approx 2.6\)):

• Option A is incorrect because the relationship described is not between the number of bankruptcies and the standard deviation relative to the mean in this way.
• Option B is incorrect because "always" is too absolute; data can deviate beyond the standard deviation (though it is less common).
• Option C is incorrect because the number of bankruptcies is not "always" equal to the standard deviation (the standard deviation is a measure of spread, not the actual count).
• Option D is correct. The standard deviation measures the typical spread of data around the mean. In most cases (for many distributions, including Poisson in practice), data values usually differ from the mean by no more than the standard deviation (though extreme deviations are possible but less frequent).

Answer:

D. The number of bankruptcies usually differs from the mean by no more than the standard deviation number of bankruptcies.