Refer to Exercises 4.2 and 4.13. Each home with a dust mite level that exceeds 2mg/g will spend $2,000 for an allergen air purification system. Find the mean and variance of the total amount spent by the four sampled homes. Give a range where this total is likely to fall.

Data from Exercise 4.2

A dust mite allergen level that exceeds 2 micrograms per gram (mg/g) of dust has been associated with the development of allergies. Consider a random sample of four homes and let Y be the number of homes with a dust mite level that exceeds 2 mg/g. The probability distribution for , based on a study by the National Institute of Environmental Health Sciences, is shown in the following table.

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y 1 3 4 p(y) .09 30 .37 20 .04 2.