Question

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, 0.05). (Round your probabilities to three decimal places.)

A button hyperlink to the SALT program that reads: Use SALT.
(a)
Determine both P(X ≤ 2) and P(X < 2).
P(X ≤ 2)
=
P(X < 2)
=
(b)
Determine P(X ≥ 3).
P(X ≥ 3)
=
(c)
Determine P(1 ≤ X ≤ 2).
P(1 ≤ X ≤ 2) =
(d)
What are E(X) and X? (Round your answers to two decimal places.)
E(X)
=
X
=
(e)
In a sample of 30 children, what is the probability that none has a food allergy?

Answer 1

Explanation:

(a)

P(X ≤ 2) can be calculated as follows:

P(X ≤ 2) = P(X = 0) + P(X = 1) + P(X = 2)

Using the binomial probability formula: P(X = k) = (n choose k) * p^k * (1 - p)^(n - k)

P(X = 0) = (25 choose 0) * 0.05^0 * (1 - 0.05)^(25 - 0)

P(X = 1) = (25 choose 1) * 0.05^1 * (1 - 0.05)^(25 - 1)

P(X = 2) = (25 choose 2) * 0.05^2 * (1 - 0.05)^(25 - 2)

Using a statistical software or calculator, these probabilities can be calculated. For example, using the SALT program:

P(X ≤ 2) = 0.594

P(X < 2) can be calculated as follows:

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

Using the binomial probability formula:

P(X < 2) = (25 choose 0) * 0.05^0 * (1 - 0.05)^(25 - 0) + (25 choose 1) * 0.05^1 * (1 - 0.05)^(25 - 1)

Using a statistical software or calculator:

P(X < 2) = 0.531

(b)

P(X ≥ 3) can be calculated as follows:

P(X ≥ 3) = 1 - P(X < 3)

Using the binomial probability formula:

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

Using a statistical software or calculator:

P(X ≥ 3) = 1 - 0.531 = 0.469

(c)

P(1 ≤ X ≤ 2) can be calculated as follows:

P(1 ≤ X ≤ 2) = P(X = 1) + P(X = 2)

Using the binomial probability formula:

P(X = 1) = (25 choose 1) * 0.05^1 * (1 - 0.05)^(25 - 1)

P(X = 2) = (25 choose 2) * 0.05^2 * (1 - 0.05)^(25 - 2)

Using a statistical software or calculator:

P(1 ≤ X ≤ 2) = P(X = 1) + P(X = 2)

(d)

E(X) represents the expected value of X, which can be calculated using the formula:

E(X) = n * p

Using the given values:

E(X) = 25 * 0.05

Using a calculator:

E(X) = 1.25

The value of X is the number of children in the sample who have a food allergy, which can vary depending on the sample. It can take on any integer value from 0 to 25, inclusive.

(e)

If we consider a sample of 30 children, the probability that none of them has a food allergy can be calculated using the binomial probability formula:

P(X = 0) = (30 choose 0) * 0.05^0 * (1 - 0.05)^(30 - 0)

Using a statistical software or calculator:

P(X = 0) = 0.443