Dust mite allergies. A dust mite allergen level that exceeds 2 micrograms per gram 1mg>g2 of dust has been associated with the development of allergies. Consider a random sample of four homes, and let x be the number of homes with a dust mite level that exceeds 2 mg>g. The probability distribution LO9 for x, based on a study, is shown in the following table:

x 0 1 2 3 4 p(x) .07 .31 .38 .17 .07

a. Verify that the probabilities for x in the table sum to 1.

b. Find the probability that three or four of the homes in the sample have a dust mite level that exceeds 2 mg>g.

c. Find the probability that fewer than two homes in the sample have a dust mite level that exceeds 2 mg>g.

d. Find E1x2. Give a meaningful interpretation of the result.

e. Find s.

f. Find the exact probability that x is in the interval m { 2s. Compare your answer with Chebyshev’s rule and the empirical rule.