The level of small business involvement in Australia has an impact on employment rates. As a result data collected on small business informs a wide variety of policy decisions by government.

Government figures indicate that 20% of small businesses will fail in the first year of operation.

Requirements:

(a) State the variable of interest, and suggest an appropriate model to estimate the probabilities of the possible numbers of small businesses that will fail in the first year of operation. Explain why this model is appropriate by discussing the important distinguishing features of the model.

(b) Using the model from part (a), estimate the probability that, in a randomly selected group of 12 small businesses, 2 or less will fail in the first year of operation.

(c) For a group of 12 randomly selected small businesses, find the mean number of small businesses expected to fail in the first year of operation.

(d) A random group of 65 small businesses have been selected from all small businesses that registered in December 2013. From this sample of 65, estimate the mean and standard deviation of the number of small businesses who will fail in their first year of operation.

(e) Determine the probability that, in a random sample of 65 small businesses (selected in part (d)), 5 or less will fail. State and check any assumptions, conditions or rules of thumb that should be considered before performing the calculations to determine this probability.