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?

(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

answer:

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step-by-step explanation: