Dust mite allergies. A dust mite allergen level that exceeds 2 micrograms per gram (μg/g)...

Dust mite allergies. A dust mite allergen level that exceeds 2 micrograms per gram (μg/g) 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 μg/g. The probability distribution for x , based on a study by the National Institute of Environmental Health Sciences, is shown in the following table:

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

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

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

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d. Find E(x). Give a meaningful interpretation of the result.

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e. Find σ.

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