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20 September, 02:40

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 Y=y, based on a study by the National Institute of Environmental Health Sciences, is shown in the following table.

y 0 1 2 3 4

p (y).09.30.37.20.04

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

(b) Find the probability that fewer than three homes in the sample have a dust mite level that exceeds 2 µg/g.

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  1. 20 September, 06:04
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    (a) 0.24

    (b) 0.76

    Step-by-step explanation:

    We are given that let Y be the number of homes with a dust mite level that exceeds 2 mg/g. The probability distribution for Y=y is shown below;

    y P (y)

    0 0.09

    1 0.30

    2 0.37

    3 0.20

    4 0.04

    (a) Probability that three or four of the homes in the sample have a dust mite level that exceeds 2 µg/g = P (y = 3) + P (y = 4)

    = 0.20 + 0.04 = 0.24.

    (b) Probability that fewer than three homes in the sample have a dust mite level that exceeds 2 µg/g = P (y = 0) + P (y = 1) + P (y = 2)

    = 0.09 + 0.30 + 0.37 = 0.76.
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