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6 May, 01:46

NBC News reported on May 2, 2013, that 1 in 20 children in the United States have a food allergy of some sort. Consider selecting a random sample of 25 children and let X be the number in the sample who have a food allergy. Then X ~ Bin (25,.05).

a. Determine both P (X < = 3) and P (X < 3).

b. Determine P (X > = 4). c. Determine P (1 < = X < = 3).

d. What are E (X) and Sigma (x) ?

e. In a sample of 50 children, what is the probablity that none have a food allergy?

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  1. 6 May, 04:33
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    (a) P (X<=3) = 0.9658, P (X<3) = 0.8728

    (b) P (X>=4) = 0.0342

    (c) P (1< = X <=3) = 0.6884

    (d) E (X) = 1.25, σ (X) = 1.089

    (e) P (X=0) = 0.0769

    Step-by-step explanation:

    We are given that X ~ Bin (25,.05) which means that this can be approximated as a binomial distribution. The formula for calculating probability using the binomial distribution is:

    P (X=x) = ⁿCₓ pˣ qⁿ⁻ˣ

    where n = total no. of trials

    x = no. of successful trials

    p = probability of success

    q = probability of failure = 1 - p

    We have n = 25, p = 0.05 and q = 0.95

    (a) P (X<=3) = P (X=0) + P (X=1) + P (X=2) + P (X=3)

    = ²⁵C₀ (0.05) ⁰ (0.95) ²⁵⁻⁰ + ²⁵C₁ (0.05) ¹ (0.95) ²⁵⁻¹ + ²⁵C₂ (0.05) ² (0.95) ²⁵⁻² + ²⁵C₃ (0.05) ³ (0.95) ²⁵⁻³

    = 0.2774 + 0.3649 + 0.2305 + 0.09302

    P (X<=3) = 0.9658

    P (X<3) = P (X=0) + P (X=1) + P (X=2)

    = ²⁵C₀ (0.05) ⁰ (0.95) ²⁵⁻⁰ + ²⁵C₁ (0.05) ¹ (0.95) ²⁵⁻¹ + ²⁵C₂ (0.05) ² (0.95) ²⁵⁻²

    = 0.2774 + 0.3649 + 0.2305

    P (X<3) = 0.8728

    (b) P (X>=4) = 1 - P (X<4)

    = 1 - (P (X=0) + P (X=1) + P (X=2) + P (X=3))

    = 1 - 0.9658

    P (X>=4) = 0.0342

    (c) P (1< = X <=3) = P (X=1) + P (X=2) + P (X=3)

    = ²⁵C₁ (0.05) ¹ (0.95) ²⁵⁻¹ + ²⁵C₂ (0.05) ² (0.95) ²⁵⁻² + ²⁵C₃ (0.05) ³ (0.95) ²⁵⁻³

    = 0.3649 + 0.2305 + 0.09302

    P (1< = X <=3) = 0.6884

    (d) E (X) = np

    = (25) (0.05)

    E (X) = 1.25

    σ (X) = √npq

    = √ (25) (0.05) (0.95)

    σ (X) = 1.089

    (e) n=50 and p = 0.05. We need P (X=0) so,

    P (X=0) = ⁵⁰C₀ (0.05) ⁰ (0.95) ⁵⁰⁻⁰

    = (1) * (1) * (0.0769)

    P (X=0) = 0.0769
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